how to determine hypothesis test statistic

Hypothesis Testing

About hypothesis testing.

Contents (Click to skip to the section):

What is a Hypothesis?

What is hypothesis testing.

• Hypothesis Testing Examples (One Sample Z Test).
• Hypothesis Test on a Mean (TI 83).

Bayesian Hypothesis Testing.

• More Hypothesis Testing Articles
• Hypothesis Tests in One Picture
• Critical Values

What is the Null Hypothesis?

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A hypothesis is an educated guess about something in the world around you. It should be testable, either by experiment or observation. For example:

• A new medicine you think might work.
• A way of teaching you think might be better.
• A possible location of new species.
• A fairer way to administer standardized tests.

It can really be anything at all as long as you can put it to the test.

What is a Hypothesis Statement?

If you are going to propose a hypothesis, it’s customary to write a statement. Your statement will look like this: “If I…(do this to an independent variable )….then (this will happen to the dependent variable ).” For example:

• If I (decrease the amount of water given to herbs) then (the herbs will increase in size).
• If I (give patients counseling in addition to medication) then (their overall depression scale will decrease).
• If I (give exams at noon instead of 7) then (student test scores will improve).
• If I (look in this certain location) then (I am more likely to find new species).

A good hypothesis statement should:

• Include an “if” and “then” statement (according to the University of California).
• Include both the independent and dependent variables.
• Be testable by experiment, survey or other scientifically sound technique.
• Be based on information in prior research (either yours or someone else’s).
• Have design criteria (for engineering or programming projects).

Hypothesis testing can be one of the most confusing aspects for students, mostly because before you can even perform a test, you have to know what your null hypothesis is. Often, those tricky word problems that you are faced with can be difficult to decipher. But it’s easier than you think; all you need to do is:

• Figure out your null hypothesis,
• State your null hypothesis,
• Choose what kind of test you need to perform,
• Either support or reject the null hypothesis .

If you trace back the history of science, the null hypothesis is always the accepted fact. Simple examples of null hypotheses that are generally accepted as being true are:

• DNA is shaped like a double helix.
• There are 8 planets in the solar system (excluding Pluto).
• Taking Vioxx can increase your risk of heart problems (a drug now taken off the market).

How do I State the Null Hypothesis?

You won’t be required to actually perform a real experiment or survey in elementary statistics (or even disprove a fact like “Pluto is a planet”!), so you’ll be given word problems from real-life situations. You’ll need to figure out what your hypothesis is from the problem. This can be a little trickier than just figuring out what the accepted fact is. With word problems, you are looking to find a fact that is nullifiable (i.e. something you can reject).

Hypothesis Testing Examples #1: Basic Example

A researcher thinks that if knee surgery patients go to physical therapy twice a week (instead of 3 times), their recovery period will be longer. Average recovery times for knee surgery patients is 8.2 weeks.

The hypothesis statement in this question is that the researcher believes the average recovery time is more than 8.2 weeks. It can be written in mathematical terms as: H 1 : μ > 8.2

Next, you’ll need to state the null hypothesis .  That’s what will happen if the researcher is wrong . In the above example, if the researcher is wrong then the recovery time is less than or equal to 8.2 weeks. In math, that’s: H 0 μ ≤ 8.2

Rejecting the null hypothesis

Ten or so years ago, we believed that there were 9 planets in the solar system. Pluto was demoted as a planet in 2006. The null hypothesis of “Pluto is a planet” was replaced by “Pluto is not a planet.” Of course, rejecting the null hypothesis isn’t always that easy— the hard part is usually figuring out what your null hypothesis is in the first place.

Hypothesis Testing Examples (One Sample Z Test)

The one sample z test isn’t used very often (because we rarely know the actual population standard deviation ). However, it’s a good idea to understand how it works as it’s one of the simplest tests you can perform in hypothesis testing. In English class you got to learn the basics (like grammar and spelling) before you could write a story; think of one sample z tests as the foundation for understanding more complex hypothesis testing. This page contains two hypothesis testing examples for one sample z-tests .

One Sample Hypothesis Testing Example: One Tailed Z Test

A principal at a certain school claims that the students in his school are above average intelligence. A random sample of thirty students IQ scores have a mean score of 112.5. Is there sufficient evidence to support the principal’s claim? The mean population IQ is 100 with a standard deviation of 15.

Step 1: State the Null hypothesis . The accepted fact is that the population mean is 100, so: H 0 : μ = 100.

Step 2: State the Alternate Hypothesis . The claim is that the students have above average IQ scores, so: H 1 : μ > 100. The fact that we are looking for scores “greater than” a certain point means that this is a one-tailed test.

Step 4: State the alpha level . If you aren’t given an alpha level , use 5% (0.05).

Step 5: Find the rejection region area (given by your alpha level above) from the z-table . An area of .05 is equal to a z-score of 1.645.

Step 6: If Step 6 is greater than Step 5, reject the null hypothesis. If it’s less than Step 5, you cannot reject the null hypothesis. In this case, it is more (4.56 > 1.645), so you can reject the null.

One Sample Hypothesis Testing Examples: #3

Blood glucose levels for obese patients have a mean of 100 with a standard deviation of 15. A researcher thinks that a diet high in raw cornstarch will have a positive or negative effect on blood glucose levels. A sample of 30 patients who have tried the raw cornstarch diet have a mean glucose level of 140. Test the hypothesis that the raw cornstarch had an effect.

• State the null hypothesis : H 0 :μ=100
• State the alternate hypothesis : H 1 :≠100
• State your alpha level. We’ll use 0.05 for this example. As this is a two-tailed test, split the alpha into two. 0.05/2=0.025
• Find the z-score associated with your alpha level . You’re looking for the area in one tail only . A z-score for 0.75(1-0.025=0.975) is 1.96. As this is a two-tailed test, you would also be considering the left tail (z = 1.96)
•   If Step 5 is less than -1.96 or greater than 1.96 (Step 3), reject the null hypothesis . In this case, it is greater, so you can reject the null.

*This process is made much easier if you use a TI-83 or Excel to calculate the z-score (the “critical value”). See:

• Critical z value TI 83
• Z Score in Excel

Hypothesis Testing Examples: Mean (Using TI 83)

You can use the TI 83 calculator for hypothesis testing, but the calculator won’t figure out the null and alternate hypotheses; that’s up to you to read the question and input it into the calculator.

Example problem : A sample of 200 people has a mean age of 21 with a population standard deviation (σ) of 5. Test the hypothesis that the population mean is 18.9 at α = 0.05.

Step 1: State the null hypothesis. In this case, the null hypothesis is that the population mean is 18.9, so we write: H 0 : μ = 18.9

Step 2: State the alternative hypothesis. We want to know if our sample, which has a mean of 21 instead of 18.9, really is different from the population, therefore our alternate hypothesis: H 1 : μ ≠ 18.9

Step 3: Press Stat then press the right arrow twice to select TESTS.

Step 4: Press 1 to select 1:Z-Test… . Press ENTER.

Step 5: Use the right arrow to select Stats .

Step 6: Enter the data from the problem: μ 0 : 18.9 σ: 5 x : 21 n: 200 μ: ≠μ 0

Step 7: Arrow down to Calculate and press ENTER. The calculator shows the p-value: p = 2.87 × 10 -9

This is smaller than our alpha value of .05. That means we should reject the null hypothesis .

Bayesian Hypothesis Testing: What is it?

Bayesian hypothesis testing helps to answer the question: Can the results from a test or survey be repeated? Why do we care if a test can be repeated? Let’s say twenty people in the same village came down with leukemia. A group of researchers find that cell-phone towers are to blame. However, a second study found that cell-phone towers had nothing to do with the cancer cluster in the village. In fact, they found that the cancers were completely random. If that sounds impossible, it actually can happen! Clusters of cancer can happen simply by chance . There could be many reasons why the first study was faulty. One of the main reasons could be that they just didn’t take into account that sometimes things happen randomly and we just don’t know why.

It’s good science to let people know if your study results are solid, or if they could have happened by chance. The usual way of doing this is to test your results with a p-value . A p value is a number that you get by running a hypothesis test on your data. A P value of 0.05 (5%) or less is usually enough to claim that your results are repeatable. However, there’s another way to test the validity of your results: Bayesian Hypothesis testing. This type of testing gives you another way to test the strength of your results.

Traditional testing (the type you probably came across in elementary stats or AP stats) is called Non-Bayesian. It is how often an outcome happens over repeated runs of the experiment. It’s an objective view of whether an experiment is repeatable. Bayesian hypothesis testing is a subjective view of the same thing. It takes into account how much faith you have in your results. In other words, would you wager money on the outcome of your experiment?

Differences Between Traditional and Bayesian Hypothesis Testing.

Traditional testing (Non Bayesian) requires you to repeat sampling over and over, while Bayesian testing does not. The main different between the two is in the first step of testing: stating a probability model. In Bayesian testing you add prior knowledge to this step. It also requires use of a posterior probability , which is the conditional probability given to a random event after all the evidence is considered.

Arguments for Bayesian Testing.

Many researchers think that it is a better alternative to traditional testing, because it:

• Includes prior knowledge about the data.
• Takes into account personal beliefs about the results.

Arguments against.

• Including prior data or knowledge isn’t justifiable.
• It is difficult to calculate compared to non-Bayesian testing.

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Hypothesis Testing Articles

• What is Ad Hoc Testing?
• Composite Hypothesis Test
• What is a Rejection Region?
• What is a Two Tailed Test?
• How to Decide if a Hypothesis Test is a One Tailed Test or a Two Tailed Test.
• How to Decide if a Hypothesis is a Left Tailed Test or a Right-Tailed Test.
• How to State the Null Hypothesis in Statistics.
• How to Find a Critical Value .
• How to Support or Reject a Null Hypothesis.

Specific Tests:

• Brunner Munzel Test (Generalized Wilcoxon Test).
• Chi Square Test for Normality.
• Cochran-Mantel-Haenszel Test.
• Granger Causality Test .
• Hotelling’s T-Squared.
• KPSS Test .
• What is a Likelihood-Ratio Test?
• Log rank test .
• MANCOVA Assumptions.
• MANCOVA Sample Size.
• Marascuilo Procedure
• Rao’s Spacing Test
• Rayleigh test of uniformity.
• Sequential Probability Ratio Test.
• How to Run a Sign Test.
• T Test: one sample.
• T-Test: Two sample .
• Welch’s ANOVA .
• Welch’s Test for Unequal Variances .
• Z-Test: one sample .
• Z Test: Two Proportion.
• Wald Test .

Related Articles:

• What is an Acceptance Region?
• How to Calculate Chebyshev’s Theorem.
• Contrast Analysis
• Decision Rule.
• Degrees of Freedom .
• Directional Test
• False Discovery Rate
• How to calculate the Least Significant Difference.
• Levels in Statistics.
• How to Calculate Margin of Error.
• Mean Difference (Difference in Means)
• The Multiple Testing Problem .
• What is the Neyman-Pearson Lemma?
• What is an Omnibus Test?
• One Sample Median Test .
• How to Find a Sample Size (General Instructions).
• Sig 2(Tailed) meaning in results
• What is a Standardized Test Statistic?
• How to Find Standard Error
• Standardized values: Example.
• How to Calculate a T-Score.
• T-Score Vs. a Z.Score.
• Testing a Single Mean.
• Unequal Sample Sizes.
• Uniformly Most Powerful Tests.
• How to Calculate a Z-Score.

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Lesson 10 of 24 By Avijeet Biswal

Table of Contents

In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

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What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps of Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

• Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
• Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. This data should be representative of the population to infer conclusions accurately.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

• If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
• If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

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Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales.  If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

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Why Is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

• Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
• Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
• Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
• Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

When Did Hypothesis Testing Begin?

Hypothesis testing as a formalized process began in the early 20th century, primarily through the work of statisticians such as Ronald A. Fisher, Jerzy Neyman, and Egon Pearson. The development of hypothesis testing is closely tied to the evolution of statistical methods during this period.

• Ronald A. Fisher (1920s): Fisher was one of the key figures in developing the foundation for modern statistical science. In the 1920s, he introduced the concept of the null hypothesis in his book "Statistical Methods for Research Workers" (1925). Fisher also developed significance testing to examine the likelihood of observing the collected data if the null hypothesis were true. He introduced p-values to determine the significance of the observed results.
• Neyman-Pearson Framework (1930s): Jerzy Neyman and Egon Pearson built on Fisher’s work and formalized the process of hypothesis testing even further. In the 1930s, they introduced the concepts of Type I and Type II errors and developed a decision-making framework widely used in hypothesis testing today. Their approach emphasized the balance between these errors and introduced the concepts of the power of a test and the alternative hypothesis.

The dialogue between Fisher's and Neyman-Pearson's approaches shaped the methods and philosophy of statistical hypothesis testing used today. Fisher emphasized the evidential interpretation of the p-value. At the same time, Neyman and Pearson advocated for a decision-theoretical approach in which hypotheses are either accepted or rejected based on pre-determined significance levels and power considerations.

The application and methodology of hypothesis testing have since become a cornerstone of statistical analysis across various scientific disciplines, marking a significant statistical development.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

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After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

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If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 2 types of hypothesis testing?

• One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
• Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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About the Author

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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Test statistics | Definition, Interpretation, and Examples

Published on July 17, 2020 by Rebecca Bevans . Revised on June 22, 2023.

The test statistic is a number calculated from a statistical test of a hypothesis. It shows how closely your observed data match the distribution expected under the null hypothesis of that statistical test.

The test statistic is used to calculate the p value of your results, helping to decide whether to reject your null hypothesis.

Table of contents

What exactly is a test statistic, types of test statistics, interpreting test statistics, reporting test statistics, other interesting articles, frequently asked questions about test statistics.

A test statistic describes how closely the distribution of your data matches the distribution predicted under the null hypothesis of the statistical test you are using.

The distribution of data is how often each observation occurs, and can be described by its central tendency and variation around that central tendency. Different statistical tests predict different types of distributions, so it’s important to choose the right statistical test for your hypothesis.

The test statistic summarizes your observed data into a single number using the central tendency, variation, sample size, and number of predictor variables in your statistical model.

Generally, the test statistic is calculated as the pattern in your data (i.e., the correlation between variables or difference between groups) divided by the variance in the data (i.e., the standard deviation ).

• Null hypothesis ( H 0 ): There is no correlation between temperature and flowering date.
• Alternate hypothesis ( H A or H 1 ): There is a correlation between temperature and flowering date.

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Below is a summary of the most common test statistics, their hypotheses, and the types of statistical tests that use them.

Different statistical tests will have slightly different ways of calculating these test statistics, but the underlying hypotheses and interpretations of the test statistic stay the same.

In practice, you will almost always calculate your test statistic using a statistical program (R, SPSS, Excel, etc.), which will also calculate the p value of the test statistic. However, formulas to calculate these statistics by hand can be found online.

• a regression coefficient of 0.36
• a t value comparing that coefficient to the predicted range of regression coefficients under the null hypothesis of no relationship

The t value of the regression test is 2.36 – this is your test statistic.

For any combination of sample sizes and number of predictor variables, a statistical test will produce a predicted distribution for the test statistic. This shows the most likely range of values that will occur if your data follows the null hypothesis of the statistical test.

The more extreme your test statistic – the further to the edge of the range of predicted test values it is – the less likely it is that your data could have been generated under the null hypothesis of that statistical test.

The agreement between your calculated test statistic and the predicted values is described by the p value . The smaller the p value, the less likely your test statistic is to have occurred under the null hypothesis of the statistical test.

Because the test statistic is generated from your observed data, this ultimately means that the smaller the p value, the less likely it is that your data could have occurred if the null hypothesis was true.

Test statistics can be reported in the results section of your research paper along with the sample size, p value of the test, and any characteristics of your data that will help to put these results into context.

Whether or not you need to report the test statistic depends on the type of test you are reporting.

By surveying a random subset of 100 trees over 25 years we found a statistically significant ( p < 0.01) positive correlation between temperature and flowering dates ( R 2 = 0.36, SD = 0.057).

In our comparison of mouse diet A and mouse diet B, we found that the lifespan on diet A  ( M = 2.1 years; SD = 0.12) was significantly shorter than the lifespan on diet B ( M = 2.6 years; SD = 0.1), with an average difference of 6 months ( t (80) = -12.75; p < 0.01).

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Confidence interval
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

A test statistic is a number calculated by a  statistical test . It describes how far your observed data is from the  null hypothesis  of no relationship between  variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

The formula for the test statistic depends on the statistical test being used.

Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation ).

The test statistic you use will be determined by the statistical test.

You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.

The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.

For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis , even if the true correlation between two variables is the same in either data set.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

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Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter .

For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

The Two Types of Statistical Hypotheses

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.

Hypothesis Tests

A hypothesis test consists of five steps:

1. State the hypotheses. 

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

2. Determine a significance level to use for the hypothesis.

Decide on a significance level. Common choices are .01, .05, and .1. 

3. Find the test statistic.

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

4. Reject or fail to reject the null hypothesis.

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The p-value  tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

5. Interpret the results. 

Interpret the results of the hypothesis test in the context of the question being asked. 

The Two Types of Decision Errors

There are two types of decision errors that one can make when doing a hypothesis test:

Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called  alpha , and denoted as α.

Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or  Beta , denoted as β.

One-Tailed and Two-Tailed Tests

A statistical hypothesis can be one-tailed or two-tailed.

A one-tailed hypothesis involves making a “greater than” or “less than ” statement.

For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.

For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

Related:   What is a Directional Hypothesis?

Types of Hypothesis Tests

There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.

The following tutorials provide an explanation of the most common types of hypothesis tests:

Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test

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Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing.

Hypothesis testing is a formal way of checking if a hypothesis about a population is true or not.

Hypothesis Testing

A hypothesis is a claim about a population parameter .

A hypothesis test is a formal procedure to check if a hypothesis is true or not.

Examples of claims that can be checked:

The average height of people in Denmark is more than 170 cm.

The share of left handed people in Australia is not 10%.

The average income of dentists is less the average income of lawyers.

The Null and Alternative Hypothesis

Hypothesis testing is based on making two different claims about a population parameter.

The null hypothesis (\(H_{0} \)) and the alternative hypothesis (\(H_{1}\)) are the claims.

The two claims needs to be mutually exclusive , meaning only one of them can be true.

The alternative hypothesis is typically what we are trying to prove.

For example, we want to check the following claim:

"The average height of people in Denmark is more than 170 cm."

In this case, the parameter is the average height of people in Denmark (\(\mu\)).

The null and alternative hypothesis would be:

Null hypothesis : The average height of people in Denmark is 170 cm.

Alternative hypothesis : The average height of people in Denmark is more than 170 cm.

The claims are often expressed with symbols like this:

\(H_{0}\): \(\mu = 170 \: cm \)

\(H_{1}\): \(\mu > 170 \: cm \)

If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

If the data does not support the alternative hypothesis, we keep the null hypothesis.

Note: The alternative hypothesis is also referred to as (\(H_{A} \)).

The Significance Level

The significance level (\(\alpha\)) is the uncertainty we accept when rejecting the null hypothesis in the hypothesis test.

The significance level is a percentage probability of accidentally making the wrong conclusion.

Typical significance levels are:

• \(\alpha = 0.1\) (10%)
• \(\alpha = 0.05\) (5%)
• \(\alpha = 0.01\) (1%)

A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.

There is no "correct" significance level - it only states the uncertainty of the conclusion.

Note: A 5% significance level means that when we reject a null hypothesis:

We expect to reject a true null hypothesis 5 out of 100 times.

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The Test Statistic

The test statistic is used to decide the outcome of the hypothesis test.

The test statistic is a standardized value calculated from the sample.

Standardization means converting a statistic to a well known probability distribution .

The type of probability distribution depends on the type of test.

Common examples are:

• Standard Normal Distribution (Z): used for Testing Population Proportions
• Student's T-Distribution (T): used for Testing Population Means

Note: You will learn how to calculate the test statistic for each type of test in the following chapters.

The Critical Value and P-Value Approach

There are two main approaches used for hypothesis tests:

• The critical value approach compares the test statistic with the critical value of the significance level.
• The p-value approach compares the p-value of the test statistic and with the significance level.

The Critical Value Approach

The critical value approach checks if the test statistic is in the rejection region .

The rejection region is an area of probability in the tails of the distribution.

The size of the rejection region is decided by the significance level (\(\alpha\)).

The value that separates the rejection region from the rest is called the critical value .

Here is a graphical illustration:

If the test statistic is inside this rejection region, the null hypothesis is rejected .

For example, if the test statistic is 2.3 and the critical value is 2 for a significance level (\(\alpha = 0.05\)):

We reject the null hypothesis (\(H_{0} \)) at 0.05 significance level (\(\alpha\))

The P-Value Approach

The p-value approach checks if the p-value of the test statistic is smaller than the significance level (\(\alpha\)).

The p-value of the test statistic is the area of probability in the tails of the distribution from the value of the test statistic.

If the p-value is smaller than the significance level, the null hypothesis is rejected .

The p-value directly tells us the lowest significance level where we can reject the null hypothesis.

For example, if the p-value is 0.03:

We reject the null hypothesis (\(H_{0} \)) at a 0.05 significance level (\(\alpha\))

We keep the null hypothesis (\(H_{0}\)) at a 0.01 significance level (\(\alpha\))

Note: The two approaches are only different in how they present the conclusion.

Steps for a Hypothesis Test

The following steps are used for a hypothesis test:

• Check the conditions
• Define the claims
• Decide the significance level
• Calculate the test statistic

One condition is that the sample is randomly selected from the population.

The other conditions depends on what type of parameter you are testing the hypothesis for.

Common parameters to test hypotheses are:

• Proportions (for qualitative data)
• Mean values (for numerical data)

You will learn the steps for both types in the following pages.

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Significance levels: what, why, and how, the statsig team.

In a world where data-driven decisions reign supreme, understanding statistical significance is like having a trusty compass to navigate the vast ocean of information. Just as a compass guides sailors to their destination, statistical significance helps researchers and analysts separate meaningful insights from random noise, ensuring they're on the right course.

Statistical significance is a crucial concept in data analysis, acting as a gatekeeper between coincidence and genuine patterns. It's the key to unlocking the true potential of your data, enabling you to make informed decisions with confidence.

Understanding significance levels

Statistical significance is a measure of the reliability and trustworthiness of your data analysis results. It helps you determine whether the patterns or differences you observe in your data are likely to be real or just a result of random chance.

Significance levels play a central role in hypothesis testing , a process used to make data-driven decisions. When you conduct a hypothesis test, you start with a null hypothesis (usually assuming no effect or difference) and an alternative hypothesis (proposing an effect or difference exists). The significance level you choose (commonly denoted as α) sets the threshold for rejecting the null hypothesis.

For example, if you set a significance level of 0.05 (5%), you're essentially saying, "I'm willing to accept a 5% chance of rejecting the null hypothesis when it's actually true." This means that if your p-value (the probability of observing results as extreme as yours, assuming the null hypothesis is true) is less than 0.05, you can confidently reject the null hypothesis and conclude that your results are statistically significant.

However, it's crucial to understand that p-values are often misinterpreted . A common misconception is that a p-value tells you the probability that your null hypothesis is true. In reality, it only tells you the probability of observing results as extreme as yours if the null hypothesis were true.

Another misinterpretation is that a smaller p-value always implies a larger effect size or practical importance. While a small p-value suggests that your results are unlikely to be due to chance, it doesn't necessarily mean that the effect is large or practically meaningful.

To find the appropriate significance level for your analysis, consider factors such as:

The consequences of making a Type I error (false positive) or Type II error (false negative)

The sample size and expected effect size

The conventions in your field of study

By carefully selecting your significance level and interpreting your p-values correctly, you can make sound decisions based on your data analysis results. Remember, statistical significance is just one piece of the puzzle – always consider the practical implications and context of your findings to make truly meaningful conclusions.

Why significance levels matter

Significance levels are crucial for distinguishing meaningful patterns from random noise in data. They help businesses avoid making decisions based on chance fluctuations. Setting the right significance level ensures that resources are allocated to genuine insights.

Significance levels impact business decisions and resource allocation . A stringent significance level (e.g., 0.01) reduces false positives but may miss valuable insights. A relaxed level (e.g., 0.10) captures more potential effects but risks acting on false positives. Choosing the appropriate level depends on the cost of false positives versus false negatives for your business.

Balancing statistical significance with practical relevance is key in real-world applications . A statistically significant result may not have a meaningful impact on user experience or revenue. When deciding how to find significance level, consider the practical implications alongside the statistical evidence . Focus on changes that drive tangible improvements for your users and business.

Calculating statistical significance

Formulating hypotheses is the first step in calculating statistical significance . Start by defining a null hypothesis (no significant difference) and an alternative hypothesis (presence of a meaningful difference). Choose a significance level , typically 0.01 or 0.05, which represents the probability of rejecting the null hypothesis when it's true.

Statistical tests help determine if observed differences are statistically significant. T-tests compare means between two groups, while chi-square tests analyze categorical data. ANOVA (Analysis of Variance) compares means among three or more groups. The choice of test depends on your data type and experimental design .

P-values indicate the probability of obtaining observed results if the null hypothesis is true. Compare the p-value to your chosen significance level to determine statistical significance. If the p-value is less than or equal to the significance level, reject the null hypothesis and conclude that the results are statistically significant.

To find significance level, consider the consequences of a Type I error (false positive) and a Type II error (false negative). A lower significance level reduces the risk of a Type I error but increases the risk of a Type II error. Balance these risks based on the context and implications of your study.

Sample size plays a crucial role in determining statistical significance. Larger sample sizes increase the power of a statistical test, making it easier to detect significant differences. However, an excessively large sample size can make even minor differences statistically significant, so consider practical relevance alongside statistical significance .

Effect size measures the magnitude of a difference or relationship. It provides context for interpreting statistically significant results. A small p-value doesn't always imply a large effect size, so consider both when drawing conclusions and making decisions based on your analysis .

Common pitfalls in significance testing

Overlooking sample size can lead to false conclusions. Smaller samples have less power to detect real differences, while larger samples may flag trivial differences as significant.

Misinterpreting p-values is another common mistake. A low p-value indicates strong evidence against the null hypothesis but doesn't measure the size or importance of an effect.

External factors like seasonality, marketing campaigns, or technical issues can influence results. Failing to account for these variables can skew your analysis and lead to incorrect conclusions.

To find significance level accurately:

Clearly define your null and alternative hypotheses upfront. This helps frame your analysis and interpretation of results.

Choose an appropriate significance level (usually 0.05 or 0.01) before collecting data. Stick to this predetermined level to avoid "p-hacking" or manipulating data to achieve significance.

Use the right statistical test for your data and research question. Different tests have different assumptions and are suited for various types of data.

Interpret results in context, considering both statistical significance and practical importance. A statistically significant result may not be meaningful if the effect size is small.

Replicate findings with new data when possible. Consistent results across multiple studies strengthen evidence for a genuine effect.

By understanding these pitfalls and best practices for finding significance level, you can make more reliable inferences from your data .

Practical applications of significance testing

Significance testing is a powerful tool for making data-driven decisions across various industries. By leveraging significance levels, product teams can optimize user experiences and drive meaningful improvements. Here's how you can apply significance testing in practice:

Using significance levels in product development

Identify high-impact features : Conduct A/B tests to determine which product features significantly improve user engagement or satisfaction. Focus development efforts on features that demonstrate statistically significant improvements.

Optimize user flows : Test different user flow variations to find the most intuitive and efficient paths. Use significance levels to validate that the chosen flow outperforms alternatives.

Refine UI/UX elements : Experiment with various UI/UX elements, such as button placement, color schemes, or typography. Analyze results using significance testing to select the most effective designs.

Applying statistical significance to marketing campaigns

Evaluate ad effectiveness : Compare the performance of different ad creatives, targeting strategies, or platforms. Use significance testing to identify the most impactful approaches and allocate marketing budgets accordingly.

Optimize landing pages : Test different landing page variations to maximize conversion rates. Determine the significance level of each variation's performance to implement the most effective design.

Refine email campaigns : Experiment with subject lines, email content, and call-to-actions. Use significance testing to identify the elements that drive the highest open and click-through rates.

Leveraging significance testing for data-driven decision making

Validate business strategies : Test different pricing models, product bundles, or promotional offers. Use significance levels to determine which strategies yield the best results and align with business objectives.

Improve customer support : Experiment with various support channels, response times, or communication styles. Analyze the significance of each approach's impact on customer satisfaction and loyalty.

Optimize resource allocation : Test different resource allocation strategies across departments or projects. Use significance testing to identify the most efficient and effective approaches for maximizing ROI.

By embracing significance testing as a core part of their decision-making process, organizations can confidently optimize their products , marketing efforts, and overall strategies. Significance levels provide a clear framework for determining which ideas and approaches are worth pursuing, enabling teams to focus on the most impactful initiatives.

To find significance levels, start by defining clear hypotheses and selecting appropriate statistical tests . Collect data through well-designed experiments and analyze the results using the chosen tests. Compare the p-values obtained against the predetermined significance level (e.g., 0.05) to determine if the observed differences are statistically significant.

Remember, while significance testing is a valuable tool, it should be used in conjunction with other factors, such as practical significance, user feedback, and business goals . By combining statistical insights with a holistic understanding of your users and industry, you can make informed decisions that drive meaningful growth and success.

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S.3.2 hypothesis testing (p-value approach).

The P -value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis was true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P -value is small, say less than (or equal to) \(\alpha\), then it is "unlikely." And, if the P -value is large, say more than \(\alpha\), then it is "likely."

If the P -value is less than (or equal to) \(\alpha\), then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the P -value is greater than \(\alpha\), then the null hypothesis is not rejected.

Specifically, the four steps involved in using the P -value approach to conducting any hypothesis test are:

• Specify the null and alternative hypotheses.
• Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. Again, to conduct the hypothesis test for the population mean μ , we use the t -statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a t -distribution with n - 1 degrees of freedom.
• Using the known distribution of the test statistic, calculate the P -value : "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?")
• Set the significance level, \(\alpha\), the probability of making a Type I error to be small — 0.01, 0.05, or 0.10. Compare the P -value to \(\alpha\). If the P -value is less than (or equal to) \(\alpha\), reject the null hypothesis in favor of the alternative hypothesis. If the P -value is greater than \(\alpha\), do not reject the null hypothesis.

Example S.3.2.1

Mean gpa section  .

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * equaling 2.5. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right Tailed

The P -value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the probability that we would observe a test statistic greater than t * = 2.5 if the population mean \(\mu\) really were 3. Recall that probability equals the area under the probability curve. The P -value is therefore the area under a t n - 1 = t 14 curve and to the right of the test statistic t * = 2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than \(\alpha\) = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ > 3 if we lowered our willingness to make a Type I error to \(\alpha\) = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

Left Tailed

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the probability that we would observe a test statistic less than t * = -2.5 if the population mean μ really were 3. The P -value is therefore the area under a t n - 1 = t 14 curve and to the left of the test statistic t* = -2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ < 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

In our example concerning the mean grade point average, suppose again that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 is the probability that we would observe a test statistic less than -2.5 or greater than 2.5 if the population mean μ really was 3. That is, the two-tailed test requires taking into account the possibility that the test statistic could fall into either tail (hence the name "two-tailed" test). The P -value is, therefore, the area under a t n - 1 = t 14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually.

Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests. The P -value, 0.0254, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0254, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ ≠ 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0254, is then greater than \(\alpha\) = 0.01.

Now that we have reviewed the critical value and P -value approach procedures for each of the three possible hypotheses, let's look at three new examples — one of a right-tailed test, one of a left-tailed test, and one of a two-tailed test.

The good news is that, whenever possible, we will take advantage of the test statistics and P -values reported in statistical software, such as Minitab, to conduct our hypothesis tests in this course.

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Statistical Significance Calculator

Try qualtrics for free, statistical significance calculator: tool & complete guide.

18 min read When you make changes to your products or services, our statistical significance calculator helps you assess how they affect sales. Learn about statistical significance and how to calculate it in this short guide.

Conversions

Conversion Rate

Significant result!

Variant B’s conversion rate (5.20)% was higher than variant A’s conversion rate (4.33)%. You can be 95% confident that variant B will perform better than variant A.

Assuming you intented to have a 50% / 50% split, a Sample Ratio Mismatch (SRM) check indicates there might be a problem with your distribution.

What is statistical significance?

If you’re not a researcher, scientist or statistician, it’s incredibly easy to misunderstand what’s meant by statistical significance. In common parlance, significance means “important”, but when researchers say the findings of a study were or are “statistically significant”, it means something else entirely.

Put simply, statistical significance refers to whether any differences observed between groups studied are “real” or simply due to chance or coincidence. If a result is statistically significant, it means that it’s unlikely to have occurred as a result of chance or a random factor.

Even if data appears to have a strong relationship, you must account for the possibility that the apparent correlation is due to random chance or sampling error.

For example, consider you’re running a study for a new pair of running shoes designed to improve average running speed.

You have two groups, Group A and Group B. Group A received the new running shoes, while Group B did not. Over the course of a month, Group A’s average running speed increased by 2km/h — but Group B (who didn’t receive the new running shoes) also increased their average running speed by 1.5km/h.

The question is, did the running shoes produce the 0.5km/h difference between the groups, or did Group A simply increase their speed by chance? Is the result statistically significant?

How do you test for statistical significance?

In quantitative research , you analyse data using null hypothesis testing. This procedure determines whether a relationship or difference between variables is statistically significant.

• Null hypothesis: Predicts no true effect, relationship or difference between variables or groups. This test aims to support the main prediction by rejecting other explanations.
• Alternative hypothesis: States your main prediction of a true effect, relationship or difference between groups and variables. This is your initial prediction that you want to prove.

Hypothesis testing always starts with the assumption that the null hypothesis is true. With this approach, you can assess the probability of obtaining the results you’re looking for — and then accept or reject the null hypothesis.

For example, you could run a test on whether eating before bed affects the quality of sleep. To start with, you have to reform your predictions into null and alternative hypotheses:

• Null hypothesis: There’s no difference in sleep quality when eating before bed.
• Alternative hypothesis: Eating before bed affects sleep quality.

When you reject a null hypothesis that’s actually true, this is called a type I error.

From here, you collect the data from the groups involved. Every statistical test will produce a test statistic, the t value, and a corresponding p-value .

What’s the t-value?

The test statistic, or t value, is a number that describes how much you want your test results to differ from the null hypothesis. It allows you to compare the average value of two data sets and determine if they come from the same population.

What is the p-value?

It’s here where it gets more complicated with the p (probability) value. The p-value  tells you the statistical significance of a finding and operates as a threshold. In most studies, a p-value  of 0.05 or less is considered statistically significant — but you can set the threshold higher.

A higher p-value  of over 0.05 means variation is less likely, while a lower value below 0.05 suggests differences. You can calculate the difference using this formula: (1 -­ p-value )*100.

What this means is that results within that threshold (give or take) are perceived as statistically significant and therefore not a result of chance or coincidence.

The next stage is interpreting your results by comparing the p-value  to a predetermined significance level.

What is a significance level?

Now, the significance level (α) is a value that you set in advance as the threshold for statistical significance. In simple terms, it’s the probability of rejecting the null hypothesis when it’s true. For example, a significance level of 0.05% indicates a 5% risk of concluding that a difference exists when there’s no actual difference.

Lower significance levels mean you require stronger, more irrefutable evidence before rejecting the null hypothesis. Also, though they sound similar, significance level and confidence level are not the same thing. Confidence level assesses the probability that if a poll/test/survey was repeated over and over again, the result obtained would remain the same.

You use the significance level in conjunction with your p-value  to determine which hypothesis the data supports. If your p-value  is less than the significance level, you can reject the null hypothesis and conclude that the results are statistically significant.

But surely there’s an easier way to test for statistical significance?

Calculate statistical significance with ease

Our statistical significance calculator helps you to understand the importance of one variable against another, but without the need for complex equations.

What you need to know before using the tool

You need to get your variables correct. Start by defining two scenarios (or hypotheses):

• Scenario one has a control variable that indicates the ‘usual’ situation, where there is no known relationship between the metrics being looked at. This is also known as the null hypothesis, which is expected to bring little to no variation between the control variable and the tested variable. This can be verified by calculating the z score (see below).
• Scenario two has a variant variable which is used to see if there is a causal relationship present.

You can test your hypotheses by calculating the z score and p value.

What is the z score?

The z-score is the numerical representation of your desired confidence level. It tells you how many standard deviations from the mean your score is.

The most common percentages are 90%, 95%, and 99%. It’s also recommended to carry out two-sided tests — but more on that later.

To find out more about z scores and how to use them, check out our sample size calculator tool.

How does the tool calculate statistical significance?

When you’re confident in the variables you placed in your hypotheses, you’re ready to use the tool. The tool works in two stages:

• First, it calculates the impact of two metrics across the two scenarios,
• Then, it compares the two data sets to see which scenario did better, and to what extent (is there a large difference or a small difference between new flavour sales on a hot day and a cold day?).

You’ll then be left with an error-free indication of the impact of an action (e.g. eating) on a reference data set (sleep quality), while excluding other elements (mattress, weather etc). This will show researchers the extent – or significance – of the impact (if there is a large or small correlation).

This is essentially a two-sided test, which is recommended for understanding statistical significance. Unlike a one-sided test that compares one variable with another to give an out-of-context conclusion, a two-sided test adds in a sense of scale.

For example, the performance level of the variant’s impact can be negative, as well as positive. In this way, a two-sided test gives you more data to determine if the variant’s impact is a real difference or just a random chance.

Here’s another example: let’s say you launch a brand-new ice cream flavour. On the first day of marketing it to customers, you happen to have excellent sales. However, it just so happens that it was also the hottest day of the year.

How can you say with certainty that rather than the weather, the new flavour was the cause for the increase in sales revenue? Let’s add the ice cream sales data to the calculator and find out.

Insert snapshot graphic of the ice cream variables into the calculator using example data: e.g.

  May 1st (new flavour is the constant on a cold day – this is the control): Ice cream scoops sold = 50 and total sales revenue = £2500

May 2nd (new flavour is the constant on a hot day – this is the variant): Ice cream scoops sold = 51 and total sales revenue = £2505

In this case, the hot weather did not impact the number of scoops sold, so we can determine that there is almost zero chance of the hot weather affecting sales volume.

So, how do I know when something is statistically significant?

This is where the p-value  comes back into play.

Where there is a larger variation in test results (e.g. a large conversion rate) between the control and variant scenarios, this means that there is likely to be a statistically significant difference between them. If the variant scenario causes more positive impact – e.g. a surge in sales – this can indicate that the variant is more likely to cause the perceived change. It’s unlikely that this is a coincidence.

Where there is less variation in results (e.g. a small conversion rate), then there is less statistical difference, and so the variant does not have as big an impact. Where the impact is not favourable – e.g. there was little upwards growth in sales revenue – this could indicate that the variant is not the cause of the sales revenue, and is therefore unlikely to help it grow.

Did the p-value  you expected come out in the results?

Example: A/B Testing Calculator

Another example of statistical significance happens in email marketing. Most email  management systems (EMSs) have the ability to run an A/B test with a representative sample size.

An A/B test helps marketers to understand whether one change between identical emails – for example, a difference in the subject line, the inclusion of an image, and adding in the recipient’s name in the greeting to personalise the message – can enhance engagement. Engagement can come in the form of a:

• Higher open rate (by A/B testing different subject lines)
• Higher click-through conversion rate or more traffic to the website (by A/B testing different link text)
• Higher customer loyalty (by A/B testing the email that results in the fewest clicks on the unsubscribe link)

The statistical significance calculator tool can be used in this situation. An example of exploring the conversion rate of two subject lines with A/B testing this looks like:

Insert qualtrics calculator graphic, based on the below (see comment):

Why is it important for business?

There are many benefits to using this tool:

• Management can rapidly turn around on products or services that are under-performing
• Using statistical significance can help you measure the impact of different growth initiatives to increase conversions or make positive impact
• Testing is quantitative and provides factual evidence without researcher bias
• By having a confirmed causal relationship, this can give you a confidence level that supports agile changes to a product or service for the better. For example, a low confidence level that a new ice-cream flavour affects sales can support the decision to remove that flavour from the product line

Doing more with statistical significance research

Once you get your head around it, you can do a lot with statistical significance testing. For example, you can try playing with the control and variant variables to see which changes have the greatest effect on your results.

You can also use the results to support further research or establish risk levels for the company to manage.

Some technology tools can make the process easy to scale up research and make the most of historical datasets effectively. For example Qualtrics’ powerful AI machine learning engine, iQ™  in CoreXM , automatically runs the complex text and statistical analysis.

Continue the journey with our guide to conducting market research

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Analysis & Reporting

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In a hypothesis test for population proportion, you calculated the p-value is 0.01 for the test statistic, which is a correct statement of the p-value?Group of answer choicesa)The p-value indicates that it is very rare to observe a test statistics equally or more extreme when the null hypothesis is true.b)The p-value indicates that it is very likely to observe a test statistics equally or more extreme when the null hypothesis is true.c)The p-value is calculated assuming the alternative is true.

a) The p-value indicates that it is very rare to observe a test statistic s equally or more extreme when the null hypothesis is true.

A null hypothesis is a claim or presumption that there is no association between two observed variables or that observed differences between groups are the result of chance. It is frequently employed in statistical testing, and it is represented by the symbol "H0". The null hypothesis is often the inverse of the alternative hypothesis, which posits a link or difference between the variables. To reject the null hypothesis, the observed data must be statistically significant and unlikely to have occurred by chance. In scientific research, the null hypothesis is crucial because it enables researchers to assess the significance of their data and reach reliable conclusions.

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Related Questions

Need help on this question ASAP PLEASE AND THANK YOU

which class question is this

The figure shows two parallel lines cut by a transversal what is the value of x

The value of.x will be 20 and the angles will each have a value of 90 degrees.

When two rays are linked at their ends, they create an angle in geometry. The sides or arms of the angle are what are known as these rays.

When two lines meet at a point, an angle is created.

An "angle" is the measurement of the "opening" between these two rays. It is symbolized by the character. The circularity or rotation of an angle is often measured in degrees and radians.

Angles are a common occurrence in daily life.

Angles are used by engineers and architects to create highways, structures, and sports venues.

Hence, The value of.x will be 20 and the angles will each have a value of 90 degrees.

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PLEASE HELP! A survey was conducted of 371 teenagers. Thirty-five percent of the teenagers said they occasionally smoked cigarettes. A. Find the margin of error for this survey B. Write a statement about the percentage of teenagers who occasionally smoke cigarettes

a. The margin of error for this survey is of: 4.85%.

b. The 95% confidence interval for the percentage of teenagers who occasionally smoke cigarettes is of:

(30.15%, 39.85%).

Confidence intervals for proportions are obtained using the z-distribution .

The bounds of the interval are given as follows:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which the variables used to calculated these bounds are listed as follows:

The confidence level is of 95%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.

The sample size and the estimate in this problem are given as follows:

[tex]n = 371, \pi = 0.35[/tex]

The margin of error is given as follows:

[tex]M = 1.96\sqrt{\frac{0.35(0.65)}{371}} = 0.0485[/tex]

As a percentage, it is of 4.85%.

Then the bounds of the interval are given by the estimate plus/minus the margin of error, hence:

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Select ALL equations that have no solution. A 6x – 2 – 3x = 3x – 2 B 6x – (3x + 8) = 16x C 10 + 6x = 15 + 9x – 3x D 11 + 3x – 7 = 6x + 5 – 3x E 12x + 2 = 2 + 12x

The equations have no solution .

6x – 2 – 3x = 3x – 2

10 + 6x = 15 + 9x – 3x

11 + 3x – 7 = 6x + 5 – 3x .

We have determine from the given option which option has no solution .

When an equation has a solution?

An equation with no solution will be an equation when simplified, which will be a false statement .

The equation, 6x – 2 – 3x = 3x – 2

add like terms,

So we get,This equation has no solution

10+6x=15+6x

add like terms we cannot find the value of x.

Therefore the given equation has no solution .

11 + 3x – 7 = 6x + 5 – 3x.

we cannot find the value of x.

So this equation has no solution.

Therefore, The equations 6x – 2 – 3x = 3x – 2,10 + 6x = 15 + 9x – 3x and 11 + 3x – 7 = 6x + 5 – 3x have no solution .

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Can Someone Explain Sin Cos and Tan to me, My finals are coming up and I do not understand anything about them

Sine and cosine are the fundamental trigonometric functions associated to the sides of a right triangle and tangent is a derivate trigonometric function.

Let be a right triangle , whose features are introduced in the figure attached below. The side with measure y is the side opposite to the angle θ, whereas the side with measure x is the side adjacent to the angle θ. The side with measure r represents the hypotenuse of the right triangle, which can be found by means of Pythagorean theorem :

r = √(x² + y²)

Trigonometric functions are trascendent functions that relates two sides of the right triangle. There are two fundamental trigonometric functions and four derivate trigonometric functions:

Fundamental

sin θ = y / r = (Opposite side) / (Hypotenuse)

cos θ = x / r = (Adjacent side) / (Hypotenuse)

tan θ = sin θ / cos θ = y / x = (Opposite side) / (Adjacent side)

cot θ = 1 / tan θ = cos θ / sin θ = x / y = (Adjacent side) / (Opposite side)

sec θ = 1 / cos θ = r / x = (Hypotenuse) / (Adjacent side)

csc θ = 1 / sin θ = r / y = (Hypotenuse) / (Opposite side)

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a computer costs £2500 when new. Its value depreciates 12% compounded each year. How much does it cost after 3 years?​

The computer will cost £ 33512.32

the cost of the computer = 2500

Depreciation = 12%

Cost after 3 years =?

compound interest is calculated as,

A = P(1 + r/n)nt

r = 12/100 = 0.12

So, A= P(1 + r/n)nt

A = 2500 (1+.12/1)1*3

A= 2500 (1.12)³

A= 2500 * 1.404928

A = 3512.32

So, The computer will cost £ 33512.32

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A company has a monthly time series that regularly shows sales being higher in the summer months. This is an example of which​ component?.

Answer: The Seasonal? brainleist if im right

Step-by-step explanation:

(-5)+8 x 2 please help

Do multiplication first because PEMDAS -5 + 16

Then add: 11

8x2 = 16 -5+16 = 11

If you do a question like this always use PEMDAS. Do it in order.

P arenthesis

E exponents

M ultiplcation

S ubtraction

Which of the following are rational numbers? -1 4/53 47.444... 92.555... 99

All are rational except 47.444...

Simplify 5a 8c + b + 3c-9a6b. (1 point)

8c +b+3c-9a6b

( b n 6b are common)

8c + 3c-9a7b

(8c and 3c are also common)

When we reject the null and the null is true, we have a made a _________ __________.

When we reject the null and the null is true , we have a made a type I error

The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.

null hypothesis is denoted as H₀

Reject the null hypothesis when the p-value is less than or equal to your significance level . Your sample data favor the alternative hypothesis, which suggests that the effect exists in the population. When you can reject the null hypothesis, your results are statistically significant.

when the p-value is greater than your significance level, you fail to reject the null hypothesis.

Sometimes , we reject our null hypothesis even when its true

there we made a type I error in hypothesis

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Help y’all I need this by tonight, I don’t know this ahhhhhh- I gotta do thi in 9th grade ;-; , -1/9(-5/13)

The answer will be 5/117.

Let us understand the process. First, we have to do -\

1/9x{-5/13).

Then we have to then calculate -1x-5 which is equal to 5.

Again, we have to calculate 9x13 which is equal to 117 .

Therefore, the answer will be 5/117.

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Determine the coupon payments for the following three bonds: 5. 5% coupon corporate bond (paid annually), 6. 45% coupon treasury note, and a corporate zero-coupon bond maturing in 10 years. (assume $1,000 par value. ).

For the 5.5% coupon corporate bond , the coupon payment would be $55 per year. For the 6.45% coupon treasury note, the coupon payment would be $64.50 per year.

Calculations using percentages show how much a given number contributes to the total. It is computed by dividing the number by the sum and multiplying by 100. It is denoted by the sign "%." For example, if a basket contains ten apples, three of which are red, the proportion of red apples in the basket is 30% (3/10 x 100). In a wide range of disciplines, including business, statistics, and education, percentage is often utilized. Calculating interest rates, taxes, and other financial activities in the finance industry annually frequently uses percentages. The frequency of events or the likelihood of a certain outcome are both measured using this method in statistics. Calculating grades and evaluating a student's overall performance are both done using this in education.

How to solve?

For the 5.5% coupon corporate bond, the coupon payment would be 5.5% of $1,000 = $55 per year.

For the 6.45% coupon treasury note, the coupon payment would be 6.45% of $1,000 = $64.50 per year.

For the corporate zero-coupon bond maturing in 10 years, there are no coupon payments. The bond will only make a payment of $1,000 at maturity.

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To hire an accountant to prepare taxes costs a flat fee and an additional hourly rate. The amount the accountant costs can be modeled by the function A(x) = 150 + 35x, where x represents the number of hours the accountant works to prepare the taxes and 150 represents the flat fee. What is the value of A(220) and its interpretation? OA(220) = 7850; If the accountant takes 220 hours to prepare the taxes, the cost will be $7,850. OA(220) = 7850; If the accountant takes 7,850 hours to prepare the taxes, the cost will be $220. OA(220) = 2; If the accountant takes 220 hours to prepare the taxes, the cost will be $2. 2 OA(220) = 2; If the account takes 2 hours to prepare the taxes, the cost will be $220.

A(220) = 7850; If the accountant takes 220 hours to prepare the taxes , the cost will be $7,850.

Hence option (A) is correct.

What is a linear equation in one variable?

The linear equation in one variable is an equation which is expressed in the form of ax+b = 0, where a and b are two integers , and x is a variable and has only one solution .

The cost function is A(x) = 150 + 35x

We have to find the value of A(220), so put x  = 220 in the given cost function , we get

        A(x) = 150 + 35x

        A(220)= 150 + 35(220)

                    = 150 + 7700

                    = 7850 dollars

Hence, A(220) = 7850; If the accountant takes 220 hours to prepare the taxes , the cost will be $7,850.

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Alexandra grew 9 flowers with 3 seed packets. How many seed packets does Alexandra need to have a total of 18 flowers in her garden? Solve using unit rates.

6. Technology required. A 6 oz cylindrical can of tomato paste needs to have a volume of 178 cm. The current can design uses a radius of 2.75 cm and a height of 7.5 cm. Use graphing technology to find a cylindrical design that would have less surface area so each can uses less metal.

The required parameters of a smaller surface area and a smaller amount of metal per can are a radius of 3.05 cm and a height of 6.09 cm.

The measurement that indicates the size of a region on a plane or curved surface is called area. Surface area refers to the area of an open surface or the border of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina. A three-dimensional shape's surface area is the sum of all of its faces. Finding the area of each face and adding them together gives us the surface area of a form.

S=2πr²+2πrh

S=2πr²+2*(178/r)

S=2πr²+356/r

S'=4πr-356/r²

h=178/(3.14*3.05*3.05)

h=178/29.23

The dimensions needed for less surface area so each can uses less metal is radius as 3.05 cm and height as 6.09 cm.

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Solve for r -2r+15=-3(-2r-7)-6

[tex]\bf r=0[/tex]

[tex]\bf -2r+15=-3\left(-2r-7\right)-6[/tex]

Expand: *-3(-2r-7)-6 = 6r+15* :-

[tex]\bf -2r+15=6r+15[/tex]

Subtract 6r from both sides:-

[tex]\bf -2r+15-6r=6r+15-6r[/tex]

[tex]\bf -8r+15=15[/tex]

Subtract 15 from both sides:-

[tex]\bf -8r+15-15=15-15[/tex]

[tex]\bf -8r=0[/tex]

Divide both sides by -8:-

[tex]\bf \cfrac{-8r}{-8}=\cfrac{0}{-8}[/tex]

__________________

Hope this helps you!

Have a good day!

-2r+15=-3(-2r-7)-6

-2r+15= 6r+21-6

-2r+15=6r+15

-6r.     -6r

    -15.  -15

hopes this helps

9th grade math need help asp

look at where the 2 lines meet and then look at the minutes

x-5/2 + x-1/8 = x-3/4 +x-4/3 GIVING BRAINLIEST!

There are no values of x

that make the equation true.

No solution

Let point A be (2,7) and point B be (-6,-3). What point is on the segment connecting A and B such that the distance from the point to B is 4 times the distance from the point to A?

The point on the segment connecting A and B that is 4 times as far from B as it is from A is (-8, 8).

or the distance to be 4 times the distance from B as A   the segment must be divided into (4+1) = 5 segments

  so we want the point that is 1/5 of the way from A to B

x    from A   to B   is     2   to   - 6   =   -8 units       1/5 * -8    +  2  =    0.4

y    from A   to B   is     7 to    -3    =   -10 units      1/5 * -10  +  7    = 5    

~Describe the transformation, by giving the coordinate rule. 7.) (x,y) --> (______,_______)

The ratio of yellow marbles to purple marbles was 13 to 4. If there were 102 marbles in the bag, how many were yellow marbles?

The number of yellow marbles based on the given ratio will be 78.

The ratio is the division of the two numbers.

For example, a/b, where a will be the numerator and b will be the denominator .

Proportion is the relation of a variable with another. It could be direct or inverse.

Suppose, the number of yellow marbles is Y while purple marbles are P.

As per the given,

The ratio Y/P = 13/4

Total Y + P = 102

(13/4)P + P = 102

13P + 4P = 408

And, Y = 102 - 24 = 78

Hence "The number of yellow marbles based on the given ratio will be 78".

For more information about ratios and proportions,

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Find the distance between the points $A\left(13,\ 2\right)$ and $B\left(7,\ 10\right)$ . The distance between the two points is

The distance between the two points is 10 units

From the question, we have the following parameters that can be used in our computation:

A = (13, 2) and B =(7, 10)

The distance between the two points can be calculated using the following distance equation

distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

(x, y) = (13, 2) and (7, 10)

Substitute (x, y) = (13, 2) and (7, 10) in distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

distance = √[(13 - 7)² + (2 - 10)²]

distance = 10

Hence, the distance is 10 units

Read more about distance at

https://brainly.com/question/7243416

Are the polygons similar? If they are, write a similarity statement and give the scale factor. The image consists of two quadrilaterals PQSN and AGKD. The lengths of the sides PQ = 10, QS = 7.5, SN = 15, NP = 15, AG = 12, GK = 12, KD = 8 and DA = 6. Angle P is congruent to angle K. Angle S is congruent to angle A. Angle N and G are right angles. A. no B. yes; POSN ~ KDAG, scale factor: 5 : 4 C. yes; POSN ~ KDAG, scale factor: 4 : 5

The polygons (quadrilaterals) are similar by the congruency of three interior angles and the equal ratios (5 : 4) of  a pair of adjacent sides. The correct option is therefore

B. Yes; PQSN ~ AGKD, scale factor; 5 : 4

Similar polygons are polygons that have the same interior angle , and proportional dimensions.

The specified parameters indicate that we have;

Polygon PQSN and AGKD are quadrilaterals

Length of sides of polygon PQSN

Length of side PQ = 10, QS = 7.5, SN = 15, NP = 15

Length of sides of polygon AGKD

AG = 12, GK = 12, KD = 8, DA = 6

The measure of the angles are;

∠N = ∠G = 90°

The angle sum property of a quadrilateral indicates that the measure of the fourth angle of both quadrilateral are congruent

The ratio of two adjacent sides of the quadrilaterals are;

7.5/6 and 10/8

7.5/6 = 1.25

10/8 = 1.25

[tex]\dfrac{7.5}{6} =\dfrac{10}{8} =1.25[/tex]

The ratio of two adjacent sides of the quadrilaterals PQSN and AGKD are therefore the same.

The three interior angles of quadrilateral PQSN are congruent to three interior angles of quadrilateral AGKD, and the ratio of two adjacent sides from each quadrilateral are the same, therefore,  quadrilaterals are PQSN is similar to quadrilateral AGKD

The scale factor of dilation from quadrilateral PQSN to AGKD is 7.5 : 6 = 15 : 12 = 5 : 4

Learn more about similar quadrilaterals in geometry here:

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Q is the midpoint of PR. If QR = x + 9 and PR = 3x + 9, what is PR?

Therefore, as per the solution from the linear equation in one variable , the line segment PR is 9 units.

A linear equation in one variable is one that is written in the form ax + b = 0, where a, and b are real numbers and the coefficients of x and constant, i.e. a and b, are not equal to zero.

PR is a line.

Q is the midpoint of PR. So PQ = PR.

PR = 3x + 9

According to the question, verification is

X + 9 = 3x + 9

3x – x = 9 - 9

Therefore, as PQ = PR

x + 9 = 3x + 9

0 + 9 = 0(3) + 9

  PR = 3(0) + 9

  PR = 0 +  9

Therefore, as per the solution from the linear equation in one variable, The line segment PR is 9 units.

To know more about linear equations in one variable visit:

https://brainly.com/question/17139602

Therefore, as per the solution from the linear equation in one variable, the line segment PR is 9 units.

 PR = 3(0) + 9

 PR = 0 +  9

brainly.com/question/17139602

Find the value of X and 2x

x=125,2x=250

A hexagon sum of angles is 720 degrees.

The sum of the hexagon's angles is 345+3x

what is 8x8 divided by 4

16 is the answer

hope this helps

64 divided by 4 = 16

Hello Factor this with steps please

[tex](2x-3)(4x^2+6x+9)[/tex]

Given expression:

[tex]8x^3-27[/tex]

Rewrite 8 as 2³ and 27 as 3³:

[tex]\implies 2^3 \cdot x^3-3^3[/tex]

[tex]\textsf{Apply exponent rule} \quad a^b \cdot c^b=(ac)^b[/tex]

[tex]\implies (2x)^3-3^3[/tex]

[tex]\boxed{\begin{minipage}{5cm}\underline{Difference of Cubes Formula}\\\\$a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$\\\end{minipage}}[/tex]

Apply the Difference of Cubes Formula .

[tex]\implies (2x-3)((2x)^2+2x \cdot 3+3^2)[/tex]

[tex]\implies (2x-3)(4x^2+6x+9)[/tex]

I need help with this question been stuck on it for a while

The correct options are B and C, or 3x-21 and -3x21, since the inequality sign must be "flipped," or reversed, whenever you multiply or divide both sides of the inequality .

In mathematics, " inequality " refers to a relationship between two expressions or values that is not equal to each other. An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made.

Anytime you multiply or divide both sides of the inequality, you must “flip” or change the direction of the inequality sign. This means that if you had a less than sign <, it would become a greater than sign >.

The correct options are B and C that is 3x≥-21 and -3x≤21 because anytime you multiply or divide both sides of the inequality, you must “ flip ” or change the direction of the inequality sign.

To know more about inequality ,

https://brainly.com/question/12679763

What is the slope of the line? 5(y+2)=4(x-3)5(y+2)=4(x−3)

Help | Advanced Search

Statistics > Machine Learning

Title: training guarantees of neural network classification two-sample tests by kernel analysis.

Abstract: We construct and analyze a neural network two-sample test to determine whether two datasets came from the same distribution (null hypothesis) or not (alternative hypothesis). We perform time-analysis on a neural tangent kernel (NTK) two-sample test. In particular, we derive the theoretical minimum training time needed to ensure the NTK two-sample test detects a deviation-level between the datasets. Similarly, we derive the theoretical maximum training time before the NTK two-sample test detects a deviation-level. By approximating the neural network dynamics with the NTK dynamics, we extend this time-analysis to the realistic neural network two-sample test generated from time-varying training dynamics and finite training samples. A similar extension is done for the neural network two-sample test generated from time-varying training dynamics but trained on the population. To give statistical guarantees, we show that the statistical power associated with the neural network two-sample test goes to 1 as the neural network training samples and test evaluation samples go to infinity. Additionally, we prove that the training times needed to detect the same deviation-level in the null and alternative hypothesis scenarios are well-separated. Finally, we run some experiments showcasing a two-layer neural network two-sample test on a hard two-sample test problem and plot a heatmap of the statistical power of the two-sample test in relation to training time and network complexity.

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Statistics By Jim

Making statistics intuitive

How Hypothesis Tests Work: Significance Levels (Alpha) and P values

By Jim Frost 45 Comments

Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population . In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant.

You hear about results being statistically significant all of the time. But, what do significance levels, P values, and statistical significance actually represent? Why do we even need to use hypothesis tests in statistics?

In this post, I answer all of these questions. I use graphs and concepts to explain how hypothesis tests function in order to provide a more intuitive explanation. This helps you move on to understanding your statistical results.

Hypothesis Test Example Scenario

To start, I’ll demonstrate why we need to use hypothesis tests using an example.

A researcher is studying fuel expenditures for families and wants to determine if the monthly cost has changed since last year when the average was $260 per month. The researcher draws a random sample of 25 families and enters their monthly costs for this year into statistical software. You can download the CSV data file: FuelsCosts . Below are the descriptive statistics for this year.

We’ll build on this example to answer the research question and show how hypothesis tests work.

Descriptive Statistics Alone Won’t Answer the Question

The researcher collected a random sample and found that this year’s sample mean (330.6) is greater than last year’s mean (260). Why perform a hypothesis test at all? We can see that this year’s mean is higher by $70! Isn’t that different?

Regrettably, the situation isn’t as clear as you might think because we’re analyzing a sample instead of the full population. There are huge benefits when working with samples because it is usually impossible to collect data from an entire population. However, the tradeoff for working with a manageable sample is that we need to account for sample error.

The sampling error is the gap between the sample statistic and the population parameter. For our example, the sample statistic is the sample mean, which is 330.6. The population parameter is μ, or mu, which is the average of the entire population. Unfortunately, the value of the population parameter is not only unknown but usually unknowable. Learn more about Sampling Error .

We obtained a sample mean of 330.6. However, it’s conceivable that, due to sampling error, the mean of the population might be only 260. If the researcher drew another random sample, the next sample mean might be closer to 260. It’s impossible to assess this possibility by looking at only the sample mean. Hypothesis testing is a form of inferential statistics that allows us to draw conclusions about an entire population based on a representative sample. We need to use a hypothesis test to determine the likelihood of obtaining our sample mean if the population mean is 260.

Background information : The Difference between Descriptive and Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

A Sampling Distribution Determines Whether Our Sample Mean is Unlikely

It is very unlikely for any sample mean to equal the population mean because of sample error. In our case, the sample mean of 330.6 is almost definitely not equal to the population mean for fuel expenditures.

If we could obtain a substantial number of random samples and calculate the sample mean for each sample, we’d observe a broad spectrum of sample means. We’d even be able to graph the distribution of sample means from this process.

This type of distribution is called a sampling distribution. You obtain a sampling distribution by drawing many random samples of the same size from the same population. Why the heck would we do this?

Because sampling distributions allow you to determine the likelihood of obtaining your sample statistic and they’re crucial for performing hypothesis tests.

Luckily, we don’t need to go to the trouble of collecting numerous random samples! We can estimate the sampling distribution using the t-distribution, our sample size, and the variability in our sample.

We want to find out if the average fuel expenditure this year (330.6) is different from last year (260). To answer this question, we’ll graph the sampling distribution based on the assumption that the mean fuel cost for the entire population has not changed and is still 260. In statistics, we call this lack of effect, or no change, the null hypothesis . We use the null hypothesis value as the basis of comparison for our observed sample value.

Sampling distributions and t-distributions are types of probability distributions.

Related posts : Sampling Distributions and Understanding Probability Distributions

Graphing our Sample Mean in the Context of the Sampling Distribution

The graph below shows which sample means are more likely and less likely if the population mean is 260. We can place our sample mean in this distribution. This larger context helps us see how unlikely our sample mean is if the null hypothesis is true (μ = 260).

The graph displays the estimated distribution of sample means. The most likely values are near 260 because the plot assumes that this is the true population mean. However, given random sampling error, it would not be surprising to observe sample means ranging from 167 to 352. If the population mean is still 260, our observed sample mean (330.6) isn’t the most likely value, but it’s not completely implausible either.

The Role of Hypothesis Tests

The sampling distribution shows us that we are relatively unlikely to obtain a sample of 330.6 if the population mean is 260. Is our sample mean so unlikely that we can reject the notion that the population mean is 260?

In statistics, we call this rejecting the null hypothesis. If we reject the null for our example, the difference between the sample mean (330.6) and 260 is statistically significant. In other words, the sample data favor the hypothesis that the population average does not equal 260.

However, look at the sampling distribution chart again. Notice that there is no special location on the curve where you can definitively draw this conclusion. There is only a consistent decrease in the likelihood of observing sample means that are farther from the null hypothesis value. Where do we decide a sample mean is far away enough?

To answer this question, we’ll need more tools—hypothesis tests! The hypothesis testing procedure quantifies the unusualness of our sample with a probability and then compares it to an evidentiary standard. This process allows you to make an objective decision about the strength of the evidence.

We’re going to add the tools we need to make this decision to the graph—significance levels and p-values!

These tools allow us to test these two hypotheses:

• Null hypothesis: The population mean equals the null hypothesis mean (260).
• Alternative hypothesis: The population mean does not equal the null hypothesis mean (260).

Related post : Hypothesis Testing Overview

What are Significance Levels (Alpha)?

A significance level, also known as alpha or α, is an evidentiary standard that a researcher sets before the study. It defines how strongly the sample evidence must contradict the null hypothesis before you can reject the null hypothesis for the entire population. The strength of the evidence is defined by the probability of rejecting a null hypothesis that is true. In other words, it is the probability that you say there is an effect when there is no effect.

For instance, a significance level of 0.05 signifies a 5% risk of deciding that an effect exists when it does not exist.

Lower significance levels require stronger sample evidence to be able to reject the null hypothesis. For example, to be statistically significant at the 0.01 significance level requires more substantial evidence than the 0.05 significance level. However, there is a tradeoff in hypothesis tests. Lower significance levels also reduce the power of a hypothesis test to detect a difference that does exist.

The technical nature of these types of questions can make your head spin. A picture can bring these ideas to life!

To learn a more conceptual approach to significance levels, see my post about Understanding Significance Levels .

Graphing Significance Levels as Critical Regions

On the probability distribution plot, the significance level defines how far the sample value must be from the null value before we can reject the null. The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.

To represent a significance level of 0.05, I’ll shade 5% of the distribution furthest from the null value.

The two shaded regions in the graph are equidistant from the central value of the null hypothesis. Each region has a probability of 0.025, which sums to our desired total of 0.05. These shaded areas are called the critical region for a two-tailed hypothesis test.

The critical region defines sample values that are improbable enough to warrant rejecting the null hypothesis. If the null hypothesis is correct and the population mean is 260, random samples (n=25) from this population have means that fall in the critical region 5% of the time.

Our sample mean is statistically significant at the 0.05 level because it falls in the critical region.

Related posts : One-Tailed and Two-Tailed Tests Explained , What Are Critical Values? , and T-distribution Table of Critical Values

Comparing Significance Levels

Let’s redo this hypothesis test using the other common significance level of 0.01 to see how it compares.

This time the sum of the two shaded regions equals our new significance level of 0.01. The mean of our sample does not fall within with the critical region. Consequently, we fail to reject the null hypothesis. We have the same exact sample data, the same difference between the sample mean and the null hypothesis value, but a different test result.

What happened? By specifying a lower significance level, we set a higher bar for the sample evidence. As the graph shows, lower significance levels move the critical regions further away from the null value. Consequently, lower significance levels require more extreme sample means to be statistically significant.

You must set the significance level before conducting a study. You don’t want the temptation of choosing a level after the study that yields significant results. The only reason I compared the two significance levels was to illustrate the effects and explain the differing results.

The graphical version of the 1-sample t-test we created allows us to determine statistical significance without assessing the P value. Typically, you need to compare the P value to the significance level to make this determination.

Related post : Step-by-Step Instructions for How to Do t-Tests in Excel

What Are P values?

P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.

This tortuous, technical definition for P values can make your head spin. Let’s graph it!

First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.

The total probability of the two shaded regions is 0.03112. If the null hypothesis value (260) is true and you drew many random samples, you’d expect sample means to fall in the shaded regions about 3.1% of the time. In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true. That’s the P value!

Learn more about How to Find the P Value .

Using P values and Significance Levels Together

If your P value is less than or equal to your alpha level, reject the null hypothesis.

The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01. Again, in practice, you pick one significance level before the experiment and stick with it!

Using the significance level of 0.05, the sample effect is statistically significant. Our data support the alternative hypothesis, which states that the population mean doesn’t equal 260. We can conclude that mean fuel expenditures have increased since last year.

P values are very frequently misinterpreted as the probability of rejecting a null hypothesis that is actually true. This interpretation is wrong! To understand why, please read my post: How to Interpret P-values Correctly .

Discussion about Statistically Significant Results

Hypothesis tests determine whether your sample data provide sufficient evidence to reject the null hypothesis for the entire population. To perform this test, the procedure compares your sample statistic to the null value and determines whether it is sufficiently rare. “Sufficiently rare” is defined in a hypothesis test by:

• Assuming that the null hypothesis is true—the graphs center on the null value.
• The significance (alpha) level—how far out from the null value is the critical region?
• The sample statistic—is it within the critical region?

There is no special significance level that correctly determines which studies have real population effects 100% of the time. The traditional significance levels of 0.05 and 0.01 are attempts to manage the tradeoff between having a low probability of rejecting a true null hypothesis and having adequate power to detect an effect if one actually exists.

The significance level is the rate at which you incorrectly reject null hypotheses that are actually true ( type I error ). For example, for all studies that use a significance level of 0.05 and the null hypothesis is correct, you can expect 5% of them to have sample statistics that fall in the critical region. When this error occurs, you aren’t aware that the null hypothesis is correct, but you’ll reject it because the p-value is less than 0.05.

This error does not indicate that the researcher made a mistake. As the graphs show, you can observe extreme sample statistics due to sample error alone. It’s the luck of the draw!

Related posts : Statistical Significance: Definition & Meaning and Types of Errors in Hypothesis Testing

Hypothesis tests are crucial when you want to use sample data to make conclusions about a population because these tests account for sample error. Using significance levels and P values to determine when to reject the null hypothesis improves the probability that you will draw the correct conclusion.

Keep in mind that statistical significance doesn’t necessarily mean that the effect is important in a practical, real-world sense. For more information, read my post about Practical vs. Statistical Significance .

If you like this post, read the companion post: How Hypothesis Tests Work: Confidence Intervals and Confidence Levels .

You can also read my other posts that describe how other tests work:

• How t-Tests Work
• How the F-test works in ANOVA
• How Chi-Squared Tests of Independence Work

To see an alternative approach to traditional hypothesis testing that does not use probability distributions and test statistics, learn about bootstrapping in statistics !

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Reader Interactions

December 11, 2022 at 10:56 am

For very easy concept about level of significance & p-value 1.Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 5% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 5% (may be 4% or 3% or 2% even less, it is p-value) it means his results are significant. Otherwise he has error > 5% (may be 6% or 7% or 8% even more, it is p-value) it means his results are non-significant. 2. Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 1% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 1% (may be 0.9% or 0.8% or 0.7% even less, it is p-value) it means his results are significant. Otherwise he has error > 1% (may be 1.1% or 1.5% or 2% even more, it is p-value) it means his results are non-significant. p-value is significant or not mainly dependent upon the level of significance.

December 11, 2022 at 7:50 pm

I think that approach helps explain how to determine statistical significance–is the p-value less than or equal to the significance level. However, it doesn’t really explain what statistical significance means. I find that comparing the p-value to the significance level is the easy part. Knowing what it means and how to choose your significance level is the harder part!

December 3, 2022 at 5:54 pm

What would you say to someone who believes that a p-value higher than the level of significance (alpha) means the null hypothesis has been proven? Should you support that statement or deny it?

December 3, 2022 at 10:18 pm

Hi Emmanuel,

When the p-value is greater than the significance level, you fail to reject the null hypothesis . That is different than proving it. To learn why and what it means, click the link to read a post that I’ve written that will answer your question!

April 19, 2021 at 12:27 am

Thank you so much Sir

April 18, 2021 at 2:37 pm

Hi sir, your blogs are much more helpful for clearing the concepts of statistics, as a researcher I find them much more useful. I have some quarries:

1. In many research papers I have seen authors using the statement ” means or values are statically at par at p = 0.05″ when they do some pair wise comparison between the treatments (a kind of post hoc) using some value of CD (critical difference) or we can say LSD which is calculated using alpha not using p. So with this article I think this should be alpha =0.05 or 5%, not p = 0.05 earlier I thought p and alpha are same. p it self is compared with alpha 0.05. Correct me if I am wrong.

2. When we can draw a conclusion using critical value based on critical values (CV) which is based on alpha values in different tests (e.g. in F test CV is at F (0.05, t-1, error df) when alpha is 0.05 which is table value of F and is compared with F calculated for drawing the conclusion); then why we go for p values, and draw a conclusion based on p values, even many online software do not give p value, they just mention CD (LSD)

3. can you please help me in interpreting interaction in two factor analysis (Factor A X Factor b) in Anova.

Thank You so much!

(Commenting again as I have not seen my comment in comment list; don’t know why)

April 18, 2021 at 10:57 pm

Hi Himanshu,

I manually approve comments so there will be some time lag involved before they show up.

Regarding your first question, yes, you’re correct. Test results are significant at particular significance levels or alpha. They should not use p to define the significance level. You’re also correct in that you compare p to alpha.

Critical values are a different (but related) approach for determining significance. It was more common before computer analysis took off because it reduced the calculations. Using this approach in its simplest form, you only know whether a result is significant or not at the given alpha. You just determine whether the test statistic falls within a critical region to determine statistical significance or not significant. However, it is ok to supplement this type of result with the actual p-value. Knowing the precise p-value provides additional information that significant/not significant does not provide. The critical value and p-value approaches will always agree too. For more information about why the exact p-value is useful, read my post about Five Tips for Interpreting P-values .

Finally, I’ve written about two-way ANOVA in my post, How to do Two-Way ANOVA in Excel . Additionally, I write about it in my Hypothesis Testing ebook .

January 28, 2021 at 3:12 pm

Thank you for your answer, Jim, I really appreciate it. I’m taking a Coursera stats course and online learning without being able to ask questions of a real teacher is not my forte!

You’re right, I don’t think I’m ready for that calculation! However, I think I’m struggling with something far more basic, perhaps even the interpretation of the t-table? I’m just not sure how you came up with the p-value as .03112, with the 24 degrees of freedom. When I pull up a t-table and look at the 24-degrees of freedom row, I’m not sure how any of those numbers correspond with your answer? Either the single tail of 0.01556 or the combined of 0.03112. What am I not getting? (which, frankly, could be a lot!!) Again, thank you SO much for your time.

January 28, 2021 at 11:19 pm

Ah ok, I see! First, let me point you to several posts I’ve written about t-values and the t-distribution. I don’t cover those in this post because I wanted to present a simplified version that just uses the data in its regular units. The basic idea is that the hypothesis tests actually convert all your raw data down into one value for a test statistic, such as the t-value. And then it uses that test statistic to determine whether your results are statistically significant. To be significant, the t-value must exceed a critical value, which is what you lookup in the table. Although, nowadays you’d typically let your software just tell you.

So, read the following two posts, which covers several aspects of t-values and distributions. And then if you have more questions after that, you can post them. But, you’ll have a lot more information about them and probably some of your questions will be answered! T-values T-distributions

January 27, 2021 at 3:10 pm

Jim, just found your website and really appreciate your thoughtful, thorough way of explaining things. I feel very dumb, but I’m struggling with p-values and was hoping you could help me.

Here’s the section that’s getting me confused:

“First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.

** I’m good up to this point. Draw the picture, do the subtraction, shade the regions. BUT, I’m not sure how to figure out the area of the shaded region — even with a T-table. When I look at the T-table on 24 df, I’m not sure what to do with those numbers, as none of them seem to correspond in any way to what I’m looking at in the problem. In the end, I have no idea how you calculated each shaded area being 0.01556.

I feel like there’s a (very simple) step that everyone else knows how to do, but for some reason I’m missing it.

Again, dumb question, but I’d love your help clarifying that.

thank you, Sara

January 27, 2021 at 9:51 pm

That’s not a dumb question at all. I actually don’t show or explain the calculations for figuring out the area. The reason for that is the same reason why students never calculate the critical t-values for their test, instead you look them up in tables or use statistical software. The common reason for all that is because calculating these values is extremely complicated! It’s best to let software do that for you or, when looking critical values, use the tables!

The principal though is that percentage of the area under the curve equals the probability that values will fall within that range.

And then, for this example, you’d need to figure out the area under the curve for particular ranges!

January 15, 2021 at 10:57 am

HI Jim, I have a question related to Hypothesis test.. in Medical imaging, there are different way to measure signal intensity (from a tumor lesion for example). I tested for the same 100 patients 4 different ways to measure tumor captation to a injected dose. So for the 100 patients, i got 4 linear regression (relation between injected dose and measured quantity at tumor sites) = so an output of 4 equations Condition A output = -0,034308 + 0,0006602*input Condition B output = 0,0117631 + 0,0005425*input Condition C output = 0,0087871 + 0,0005563*input Condition D output = 0,001911 + 0,0006255*input

My question : i want to compare the 4 methods to find the best one (compared to others) : do Hypothesis test good to me… and if Yes, i do not find test to perform it. Can you suggest me a software. I uselly used JMP for my stats… but open to other softwares…

THank for your time G

November 16, 2020 at 5:42 am

Thank you very much for writing about this topic!

Your explanation made more sense to me about: Why we reject Null Hypothesis when p value < significance level

Kind greetings, Jalal

September 25, 2020 at 1:04 pm

Hi Jim, Your explanations are so helpful! Thank you. I wondered about your first graph. I see that the mean of the graph is 260 from the null hypothesis, and it looks like the standard deviation of the graph is about 31. Where did you get 31 from? Thank you

September 25, 2020 at 4:08 pm

Hi Michelle,

That is a great question. Very observant. And it gets to how these tests work. The hypothesis test that I’m illustrating here is the one-sample t-test. And this graph illustrates the sampling distribution for the t-test. T-tests use the t-distribution to determine the sampling distribution. For the t-distribution, you need to specify the degrees of freedom, which entirely defines the distribution (i.e., it’s the only parameter). For 1-sample t-tests, the degrees of freedom equal the number of observations minus 1. This dataset has 25 observations. Hence, the 24 DF you see in the graph.

Unlike the normal distribution, there is no standard deviation parameter. Instead, the degrees of freedom determines the spread of the curve. Typically, with t-tests, you’ll see results discussed in terms of t-values, both for your sample and for defining the critical regions. However, for this introductory example, I’ve converted the t-values into the raw data units (t-value * SE mean).

So, the standard deviation you’re seeing in the graph is a result of the spread of the underlying t-distribution that has 24 degrees of freedom and then applying the conversion from t-values to raw values.

September 10, 2020 at 8:19 am

Your blog is incredible.

I am having difficulty understanding why the phrase ‘as extreme as’ is required in the definition of p-value (“P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.”)

Why can’t P-Values simply be defined as “The probability of sample observation if the null hypothesis is correct?”

In your other blog titled ‘Interpreting P values’ you have explained p-values as “P-values indicate the believability of the devil’s advocate case that the null hypothesis is correct given the sample data”. I understand (or accept) this explanation. How does one move from this definition to one that contains the phrase ‘as extreme as’?

September 11, 2020 at 5:05 pm

Thanks so much for your kind words! I’m glad that my website has been helpful!

The key to understanding the “at least as extreme” wording lies in the probability plots for p-values. Using probability plots for continuous data, you can calculate probabilities, but only for ranges of values. I discuss this in my post about understanding probability distributions . In a nutshell, we need a range of values for these probabilities because the probabilities are derived from the area under a distribution curve. A single value just produces a line on these graphs rather than an area. Those ranges are the shaded regions in the probability plots. For p-values, the range corresponds to the “at least as extreme” wording. That’s where it comes from. We need a range to calculate a probability. We can’t use the single value of the observed effect because it doesn’t produce an area under the curve.

I hope that helps! I think this is a particularly confusing part of understanding p-values that most people don’t understand.

August 7, 2020 at 5:45 pm

Hi Jim, thanks for the post.

Could you please clarify the following excerpt from ‘Graphing Significance Levels as Critical Regions’:

“The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.”

I’m not sure if I understood this correctly. If the sample value fall in one of the shaded regions, doesn’t mean that the null hypothesis can be rejected, hence that is not correct?

August 7, 2020 at 10:23 pm

Think of it this way. There are two basic reasons for why a sample value could fall in a critical region:

• The null hypothesis is correct and random chance caused the sample value to be unusual.
• The null hypothesis is not correct.

You don’t know which one is true. Remember, just because you reject the null hypothesis it doesn’t mean the null is false. However, by using hypothesis tests to determine statistical significance, you control the chances of #1 occurring. The rate at which #1 occurs equals your significance level. On the hand, you don’t know the probability of the sample value falling in a critical region if the alternative hypothesis is correct (#2). It depends on the precise distribution for the alternative hypothesis and you usually don’t know that, which is why you’re testing the hypotheses in the first place!

I hope I answered the question you were asking. If not, feel free to ask follow up questions. Also, this ties into how to interpret p-values . It’s not exactly straightforward. Click the link to learn more.

June 4, 2020 at 6:17 am

Hi Jim, thank you very much for your answer. You helped me a lot!

June 3, 2020 at 5:23 pm

Hi, Thanks for this post. I’ve been learning a lot with you. My question is regarding to lack of fit. The p-value of my lack of fit is really low, making my lack of fit significant, meaning my model does not fit well. Is my case a “false negative”? given that my pure error is really low, making the computation of the lack of fit low. So it means my model is good. Below I show some information, that I hope helps to clarify my question.

SumSq DF MeanSq F pValue ________ __ ________ ______ __________

Total 1246.5 18 69.25 Model 1241.7 6 206.94 514.43 9.3841e-14 . Linear 1196.6 3 398.87 991.53 1.2318e-14 . Nonlinear 45.046 3 15.015 37.326 2.3092e-06 Residual 4.8274 12 0.40228 . Lack of fit 4.7388 7 0.67698 38.238 0.0004787 . Pure error 0.088521 5 0.017704

June 3, 2020 at 7:53 pm

As you say, a low p-value for a lack of fit test indicates that the model doesn’t fit your data adequately. This is a positive result for the test, which means it can’t be a “false negative.” At best, it could be a false positive, meaning that your data actually fit model well despite the low p-value.

I’d recommend graphing the residuals and looking for patterns . There is probably a relationship between variables that you’re not modeling correctly, such as curvature or interaction effects. There’s no way to diagnose the specific nature of the lack-of-fit problem by using the statistical output. You’ll need the graphs.

If there are no patterns in the residual plots, then your lack-of-fit results might be a false positive.

I hope this helps!

May 30, 2020 at 6:23 am

First of all, I have to say there are not many resources that explain a complicated topic in an easier manner.

My question is, how do we arrive at “if p value is less than alpha, we reject the null hypothesis.”

Is this covered in a separate article I could read?

Thanks Shekhar

May 25, 2020 at 12:21 pm

Hi Jim, terrific website, blog, and after this I’m ordering your book. One of my biggest challenges is nomenclature, definitions, context, and formulating the hypotheses. Here’s one I want to double-be-sure I understand: From above you write: ” These tools allow us to test these two hypotheses:

Null hypothesis: The population mean equals the null hypothesis mean (260). Alternative hypothesis: The population mean does not equal the null hypothesis mean (260). ” I keep thinking that 260 is the population mean mu, the underlying population (that we never really know exactly) and that the Null Hypothesis is comparing mu to x-bar (the sample mean of the 25 families randomly sampled w mean = sample mean = x-bar = 330.6).

So is the following incorrect, and if so, why? Null hypothesis: The population mean mu=260 equals the null hypothesis mean x-bar (330.6). Alternative hypothesis: The population mean mu=269 does not equal the null hypothesis mean x-bar (330.6).

And my thinking is that usually the formulation of null and alternative hypotheses is “test value” = “mu current of underlying population”, whereas I read the formulation on the webpage above to be the reverse.

Any comments appreciated. Many Thanks,

May 26, 2020 at 8:56 pm

The null hypothesis states that population value equals the null value. Now, I know that’s not particularly helpful! But, the null value varies based on test and context. So, in this example, we’re setting the null value aa $260, which was the mean from the previous year. So, our null hypothesis states:

Null: the population mean (mu) = 260. Alternative: the population mean ≠ 260.

These hypothesis statements are about the population parameter. For this type of one-sample analysis, the target or reference value you specify is the null hypothesis value. Additionally, you don’t include the sample estimate in these statements, which is the X-bar portion you tacked on at the end. It’s strictly about the value of the population parameter you’re testing. You don’t know the value of the underlying distribution. However, given the mutually exclusive nature of the null and alternative hypothesis, you know one or the other is correct. The null states that mu equals 260 while the alternative states that it doesn’t equal 260. The data help you decide, which brings us to . . .

However, the procedure does compare our sample data to the null hypothesis value, which is how it determines how strong our evidence is against the null hypothesis.

I hope I answered your question. If not, please let me know!

May 8, 2020 at 6:00 pm

Really using the interpretation “In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”, our head seems to tie a knot. However, doing the reverse interpretation, it is much more intuitive and easier. That is, we will observe the sample effect of at least 70.6 in about 96.9% of the time, if the null is false (that is, our hypothesis is true).

May 8, 2020 at 7:25 pm

Your phrasing really isn’t any simpler. And it has the additional misfortune of being incorrect.

What you’re essentially doing is creating a one-sided confidence interval by using the p-value from a two-sided test. That’s incorrect in two ways.

• Don’t mix and match one-sided and two-sided test results.
• Confidence levels are determine by the significance level, not p-values.

So, what you need is a two-sided 95% CI (1-alpha). You could then state the results are statistically significant and you have 95% confidence that the population effect is between X and Y. If you want a lower bound as you propose, then you’ll need to use a one-sided hypothesis test with a 95% Lower Bound. That’ll give you a different value for the lower bound than the one you use.

I like confidence intervals. As I write elsewhere, I think they’re easier to understand and provide more information than a binary test result. But, you need to use them correctly!

One other point. When you are talking about p-values, it’s always under the assumption that the null hypothesis is correct. You *never* state anything about the p-value in relation to the null being false (i.e. alternative is true). But, if you want to use the type of phrasing you suggest, use it in the context of CIs and incorporate the points I cover above.

February 10, 2020 at 11:13 am

Muchas gracias profesor por compartir sus conocimientos. Un saliud especial desde Colombia.

August 6, 2019 at 11:46 pm

i found this really helpful . also can you help me out ?

I’m a little confused Can you tell me if level of significance and pvalue are comparable or not and if they are what does it mean if pvalue < LS . Do we reject the null hypothesis or do we accept the null hypothesis ?

August 7, 2019 at 12:49 am

Hi Divyanshu,

Yes, you compare the p-value to the significance level. When the p-value is less than the significance level (alpha), your results are statistically significant and you reject the null hypothesis.

I’d suggest re-reading the “Using P values and Significance Levels Together” section near the end of this post more closely. That describes the process. The next section describes what it all means.

July 1, 2019 at 4:19 am

sure.. I will use only in my class rooms that too offline with due credits to your orginal page. I will encourage my students to visit your blog . I have purchased your eBook on Regressions….immensely useful.

July 1, 2019 at 9:52 am

Hi Narasimha, that sounds perfect. Thanks for buying my ebook as well. I’m thrilled to hear that you’ve found it to be helpful!

June 28, 2019 at 6:22 am

I have benefited a lot by your writings….Can I share the same with my students in the classroom?

June 30, 2019 at 8:44 pm

Hi Narasimha,

Yes, you can certainly share with your students. Please attribute my original page. And please don’t copy whole sections of my posts onto another webpage as that can be bad with Google! Thanks!

February 11, 2019 at 7:46 pm

Hello, great site and my apologies if the answer to the following question exists already.

I’ve always wondered why we put the sampling distribution about the null hypothesis rather than simply leave it about the observed mean. I can see mathematically we are measuring the same distance from the null and basically can draw the same conclusions.

For example we take a sample (say 50 people) we gather an observation (mean wage) estimate the standard error in that observation and so can build a sampling distribution about the observed mean. That sampling distribution contains a confidence interval, where say, i am 95% confident the true mean lies (i.e. in repeated sampling the true mean would reside within this interval 95% of the time).

When i use this for a hyp-test, am i right in saying that we place the sampling dist over the reference level simply because it’s mathematically equivalent and it just seems easier to gauge how far the observation is from 0 via t-stats or its likelihood via p-values?

It seems more natural to me to look at it the other way around. leave the sampling distribution on the observed value, and then look where the null sits…if it’s too far left or right then it is unlikely the true population parameter is what we believed it to be, because if the null were true it would only occur ~ 5% of the time in repeated samples…so perhaps we need to change our opinion.

Can i interpret a hyp-test that way? Or do i have a misconception?

February 12, 2019 at 8:25 pm

The short answer is that, yes, you can draw the interval around the sample mean instead. And, that is, in fact, how you construct confidence intervals. The distance around the null hypothesis for hypothesis tests and the distance around the sample for confidence intervals are the same distance, which is why the results will always agree as long as you use corresponding alpha levels and confidence levels (e.g., alpha 0.05 with a 95% confidence level). I write about how this works in a post about confidence intervals .

I prefer confidence intervals for a number of reasons. They’ll indicate whether you have significant results if they exclude the null value and they indicate the precision of the effect size estimate. Corresponding with what you’re saying, it’s easier to gauge how far a confidence interval is from the null value (often zero) whereas a p-value doesn’t provide that information. See Practical versus Statistical Significance .

So, you don’t have any misconception at all! Just refer to it as a confidence interval rather than a hypothesis test, but, of course, they are very closely related.

January 9, 2019 at 10:37 pm

Hi Jim, Nice Article.. I have a question… I read the Central limit theorem article before this article…

Coming to this article, During almost every hypothesis test, we draw a normal distribution curve assuming there is a sampling distribution (and then we go for test statistic, p value etc…). Do we draw a normal distribution curve for hypo tests because of the central limit theorem…

Thanks in advance, Surya

January 10, 2019 at 1:57 am

These distributions are actually the t-distribution which are different from the normal distribution. T-distributions only have one parameter–the degrees of freedom. As the DF of increases, the t-distribution tightens up. Around 25 degrees of freedom, the t-distribution approximates the normal distribution. Depending on the type of t-test, this corresponds to a sample size of 26 or 27. Similarly, the sampling distribution of the means also approximate the normal distribution at around these sample sizes. With a large enough sample size, both the t-distribution and the sample distribution converge to a normal distribution regardless (largely) of the underlying population distribution. So, yes, the central limit theorem plays a strong role in this.

It’s more accurate to say that central limit theorem causes the sampling distribution of the means to converge on the same distribution that the t-test uses, which allows you to assume that the test produces valid results. But, technically, the t-test is based on the t-distribution.

Problems can occur if the underlying distribution is non-normal and you have a small sample size. In that case, the sampling distribution of the means won’t approximate the t-distribution that the t-test uses. However, the test results will assume that it does and produce results based on that–which is why it causes problems!

November 19, 2018 at 9:15 am

Dear Jim! Thank you very much for your explanation. I need your help to understand my data. I have two samples (about 300 observations) with biased distributions. I did the ttest and obtained the p-value, which is quite small. Can I draw the conclusion that the effect size is small even when the distribution of my data is not normal? Thank you

November 19, 2018 at 9:34 am

Hi Tetyana,

First, when you say that your p-value is small and that you want to “draw the conclusion that the effect size is small,” I assume that you mean statistically significant. When the p-value is low, the null hypothesis must go! In other words, you reject the null and conclude that there is a statistically significant effect–not a small effect.

Now, back to the question at hand! Yes, When you have a sufficiently large sample-size, t-tests are robust to departures from normality. For a 2-sample t-test, you should have at least 15 samples per group, which you exceed by quite a bit. So, yes, you can reliably conclude that your results are statistically significant!

You can thank the central limit theorem! 🙂

September 10, 2018 at 12:18 am

Hello Jim, I am very sorry; I have very elementary of knowledge of stats. So, would you please explain how you got a p- value of 0.03112 in the above calculation/t-test? By looking at a chart? Would you also explain how you got the information that “you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”?

July 6, 2018 at 7:02 am

A quick question regarding your use of two-tailed critical regions in the article above: why? I mean, what is a real-world scenario that would warrant a two-tailed test of any kind (z, t, etc.)? And if there are none, why keep using the two-tailed scenario as an example, instead of the one-tailed which is both more intuitive and applicable to most if not all practical situations. Just curious, as one person attempting to educate people on stats to another (my take on the one vs. two-tailed tests can be seen here: http://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ )

Thanks, Georgi

July 6, 2018 at 12:05 pm

There’s the appropriate time and place for both one-tailed and two-tailed tests. I plan to write a post on this issue specifically, so I’ll keep my comments here brief.

So much of statistics is context sensitive. People often want concrete rules for how to do things in statistics but that’s often hard to provide because the answer depends on the context, goals, etc. The question of whether to use a one-tailed or two-tailed test falls firmly in this category of it depends.

I did read the article you wrote. I’ll say that I can see how in the context of A/B testing specifically there might be a propensity to use one-tailed tests. You only care about improvements. There’s probably not too much downside in only caring about one direction. In fact, in a post where I compare different tests and different options , I suggest using a one-tailed test for a similar type of casing involving defects. So, I’m onboard with the idea of using one-tailed tests when they’re appropriate. However, I do think that two-tailed tests should be considered the default choice and that you need good reasons to move to a one-tailed test. Again, your A/B testing area might supply those reasons on a regular basis, but I can’t make that a blanket statement for all research areas.

I think your article mischaracterizes some of the pros and cons of both types of tests. Just a couple of for instances. In a two-tailed test, you don’t have to take the same action regardless of which direction the results are significant (example below). And, yes, you can determine the direction of the effect in a two-tailed test. You simply look at the estimated effect. Is it positive or negative?

On the other hand, I do agree that one-tailed tests don’t increase the overall Type I error. However, there is a big caveat for that. In a two-tailed test, the Type I error rate is evenly split in both tails. For a one-tailed test, the overall Type I error rate does not change, but the Type I errors are redistributed so they all occur in the direction that you are interested in rather than being split between the positive and negative directions. In other words, you’ll have twice as many Type I errors in the specific direction that you’re interested in. That’s not good.

My big concerns with one-tailed tests are that it makes it easier to obtain the results that you want to obtain. And, all of the Type I errors (false positives) are in that direction too. It’s just not a good combination.

To answer your question about when you might want to use two-tailed tests, there are plenty of reasons. For one, you might want to avoid the situation I describe above. Additionally, in a lot of scientific research, the researchers truly are interested in detecting effects in either direction for the sake of science. Even in cases with a practical application, you might want to learn about effects in either direction.

For example, I was involved in a research study that looked at the effects of an exercise intervention on bone density. The idea was that it might be a good way to prevent osteoporosis. I used a two-tailed test. Obviously, we’re hoping that there was positive effect. However, we’d be very interested in knowing whether there was a negative effect too. And, this illustrates how you can have different actions based on both directions. If there was a positive effect, you can recommend that as a good approach and try to promote its use. If there’s a negative effect, you’d issue a warning to not do that intervention. You have the potential for learning both what is good and what is bad. The extra false-positives would’ve cause problems because we’d think that there’d be health benefits for participants when those benefits don’t actually exist. Also, if we had performed only a one-tailed test and didn’t obtain significant results, we’d learn that it wasn’t a positive effect, but we would not know whether it was actually detrimental or not.

Here’s when I’d say it’s OK to use a one-tailed test. Consider a one-tailed test when you’re in situation where you truly only need to know whether an effect exists in one direction, and the extra Type I errors in that direction are an acceptable risk (false positives don’t cause problems), and there’s no benefit in determining whether an effect exists in the other direction. Those conditions really restrict when one-tailed tests are the best choice. Again, those restrictions might not be relevant for your specific field, but as for the usage of statistics as a whole, they’re absolutely crucial to consider.

On the other hand, according to this article, two-tailed tests might be important in A/B testing !

March 30, 2018 at 5:29 am

Dear Sir, please confirm if there is an inadvertent mistake in interpretation as, “We can conclude that mean fuel expenditures have increased since last year.” Our null hypothesis is =260. If found significant, it implies two possibilities – both increase and decrease. Please let us know if we are mistaken here. Many Thanks!

March 30, 2018 at 9:59 am

Hi Khalid, the null hypothesis as it is defined for this test represents the mean monthly expenditure for the previous year (260). The mean expenditure for the current year is 330.6 whereas it was 260 for the previous year. Consequently, the mean has increased from 260 to 330.7 over the course of a year. The p-value indicates that this increase is statistically significant. This finding does not suggest both an increase and a decrease–just an increase. Keep in mind that a significant result prompts us to reject the null hypothesis. So, we reject the null that the mean equals 260.

Let’s explore the other possible findings to be sure that this makes sense. Suppose the sample mean had been closer to 260 and the p-value was greater than the significance level, those results would indicate that the results were not statistically significant. The conclusion that we’d draw is that we have insufficient evidence to conclude that mean fuel expenditures have changed since the previous year.

If the sample mean was less than the null hypothesis (260) and if the p-value is statistically significant, we’d concluded that mean fuel expenditures have decreased and that this decrease is statistically significant.

When you interpret the results, you have to be sure to understand what the null hypothesis represents. In this case, it represents the mean monthly expenditure for the previous year and we’re comparing this year’s mean to it–hence our sample suggests an increase.

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