how to do null hypothesis on excel

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This example teaches you how to perform a t-Test in Excel . The t-Test is used to test the null hypothesis that the means of two populations are equal.

Below you can find the study hours of 6 female students and 5 male students.

To perform a t-Test, execute the following steps.

1. First, perform an F-Test to determine if the variances of the two populations are equal. This is not the case.

2. On the Data tab, in the Analysis group, click Data Analysis.

Note: can't find the Data Analysis button? Click here to load the Analysis ToolPak add-in .

3. Select t-Test: Two-Sample Assuming Unequal Variances and click OK.

4. Click in the Variable 1 Range box and select the range A2:A7.

5. Click in the Variable 2 Range box and select the range B2:B6.

6. Click in the Hypothesized Mean Difference box and type 0 (H 0 : μ 1 - μ 2 = 0).

7. Click in the Output Range box and select cell E1.

8. Click OK.

Conclusion: We do a two-tail test (inequality). lf t Stat < -t Critical two-tail or t Stat > t Critical two-tail, we reject the null hypothesis. This is not the case, -2.365 < 1.473 < 2.365. Therefore, we do not reject the null hypothesis. The observed difference between the sample means (33 - 24.8) is not convincing enough to say that the average number of study hours between female and male students differ significantly.

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How to Do a T Test in Excel (2 Ways with Interpretation of Results)

The article will show you how to do a T Test in Excel. T-Tests are hypothesis tests that evaluate one or two groups’ means. Hypothesis tests employ sample data to infer population traits. In this lesson, we will look at the different types of T-Tests , and how to run T-Tests in Excel. We’ll go over both paired and two sample T-Tests , with detailed instructions on how to prepare your data, run the test, and interpret the findings.

Understanding how to use the T.TEST function in Excel will improve your ability to draw significant insights and make data-driven decisions, whether you’re a student, researcher, business analyst, or anybody else who works with data. Let’s say, you’re doing education research to assess the efficacy between traditional and new approaches. T-tests will guide you through providing the mean scores of students based on the approaches that they were taught. So that, you can make a decision based on the students’ performance.

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T Test.xlsx

T Test Type

There are basically two types of t-tests. They are:

• One-tailed t-test
• Two-tailed t-test

Each of them has 3 types. They are:

• Two sample equal variance
• Two sample unequal variance

We will show you the application of some of these types. The procedure of getting the results for all types of t-tests in Excel are the same. Let’s dig into some details and see how it can be done.

How to Do a T Test in Excel: 2 Effective Ways

1. using excel t.test or ttest function to do t test.

Here, we are going to show you how to determine the T Test result by using formulas. Excel has T.TEST and TTEST functions to operate t-test on different variables. Both functions work similarly. First, we will cover how to determine the t-test value of two sample variables with equal variance.

1.1 Two Sample Equal Variance T Test

In the dataset, you will see the prices of different laptops and smartphones. Here is a formula that performs a T Test on the prices of these products and returns the t-test result.

=T.TEST(B5:B14,C5:C14,2,2)

We set the 3rd argument of the function to 2 as we are doing a two tailed t-test on the dataset. The 4th argument should be 2 for a two sample equal variance t-test.

1.2 Paired T Test

Now, we are going to apply another formula to calculate the Paired T-Test . The following dataset shows the performance mark of some employees in two different criteria.

=T.TEST(C5:C13,D5:D13,2,1)

Note: The explanation of the results is described in the following sections.

2. Using Analysis Toolpak

The above tasks can be done with the Analysis Toolpak Add-in too. The Analysis Toolpak Add-in is not available in the ribbon by default. To initiate it,

• Go to the Options window first.
• Next, select Add-ins and click on the Go button beside the Manage section.
• After that, click OK .

• Thereafter, the Add-ins window will appear. Select Analysis Toolpak >> click OK again.

This Add-in will be added to the ribbon of the Data tab.

2.1 Two Sample Equal Variance T Test

We will do a two sample equal variance t-test using the Analysis Toolpak here. We used the dataset that contains the prices of laptops and smartphones. For this purpose,

• Click on the Data Analysis button from the ribbon of the Data tab.
• The Data Analysis features will appear. Select t-Test: Two Sample Assuming Equal Variances and click OK .

• After that, you need to set up the parameters for the t-test operation. Insert the Laptop and Smartphone prices as Variable 1 Range and Variable 2 Range Include the headings in the range and check Labels.
• Next, set the value of Hypothesized Mean Difference to 0 .
• Finally, select an Output option of your preference and click OK .

As we have chosen a New Worksheet for the outputs, we will see the results in a new sheet.

Now, let’s get to the discussion on the results.

Comments on Results

The output shows that the mean values for Laptops and Smartphones are 1608.85 and 1409.164 respectively. We can see from the Variances row that they are not precisely equal, but they are close enough to be assumed to have equal variances. The most relevant metric is the p-value .

The difference between means is statistically significant if the p-value is less than your significance level. Excel calculates p-values for one- and two-tailed T Tests .

One-tailed T Tests can detect only one direction of difference between means. A one-tailed test, for example, might only evaluate whether Smartphones have higher prices than Laptops . Two-tailed tests can reveal differences that are larger or smaller than. There are some other disadvantages to utilizing one-tailed testing, so I’ll continue with the conventional two-tailed results.

For our results, we’ll utilize P(T=t) two-tail, which is the p-value for the t-test’s two-tailed version. We cannot reject the null hypothesis because our p-value ( 0.095639932 ) is greater than the conventional significance level of 0.05 . The hypothesis that the population means differ is supported by our sample data. The mean price of Laptops is greater than the mean price of Smartphones’ .

The Analysis Toolpak operation also returns results for one-tailed t-test . Here, the one-tailed P value of two sample equal variance t-test is 1.734 .

2.2 Paired T Test

Similarly, you can find out the Paired t-Test result for the dataset containing employee performances. Just select the t-Test: Paired Two Samples for Mean when you open the Data Analysis window.

The result shows that the mean for the Workpace is 104 and the mean for the Efficiency is 96.56 .

The difference between means is statistically significant if the p-value is less than your significance level. For our results, we’ll utilize P(T=t) two-tail, which is the p-value for the t-test’s two-tailed version. We cannot reject the null hypothesis because our p-value ( 0.188 ) is greater than the conventional significance level of 0.05 . The hypothesis that the population means differ is supported by our sample data. In particular, the Workpace mean exceeds the Efficiency mean.

How to Interpret t-Test Results in Excel

Although we explained the results of the t-Test earlier, we didn’t show the proper interpretation. So here, I’ll show you the interpretation of the two sample equal variance t-test.

Let’s bring out the results again first.

• The mean of laptop prices = 1608.85
• The mean of smartphone prices = 1409.164

ii. Variance

• The variance of laptop prices = 77622.597
• The variance of smartphone prices = 51313.7904

iii. Observations

The number of observations for both laptops and smartphones are 10 .

iv. Pooled Variance

The samples’ average variance, calculated by pooling the variances of each sample.

The mathematical formula for this parameter is:

((No of observations of Sample 1-1)*(Variance of Sample 1) + (No of observations of Sample 2-1)*(Variance of Sample 2))/(No of observations of Sample 1 + No of observations of Sample 2 – 2)

So it becomes: ((10-1)*77622.59676+(10-1)*51313.7904)/(10+10-2) = 64468.19358

v. Hypothesized Mean Difference

We “hypothesize” that the number is the difference between the two population means. In this situation, we chose 0 because we want to see if the difference between the means of the two populations is zero.

It indicates the value of the Degrees of Freedom. Formula for this parameter is:

No of observations of Sample 1 + No of observations of Sample 2 – 2 = 10 + 10 – 2 = 18

vii. t Stat

The test statistic value of the t-Test operation.

The formula for this parameter is given below.

(Mean of Sample 1 – Mean of Sample 2)/(Square root of (Pooling Variance* (1/No of observations of Sample 1 + 1/No of observations of Sample 2)))

So it becomes: (1608.85 – 1409.164)/Sqrt(64468.19358 * (1/10 + 1/10)) = 1.758570846

viii. P(T<=t) two-tail

A two-tailed t-test’s p-value. This value can be found by entering t = 1.758570846 with 18 degrees of freedom into any T Score to P Value Calculator.

In this situation, the value of p is 0.095639932 . Because this is greater than 0.05 , we cannot reject the null hypothesis. This suggests that we lack adequate evidence to conclude that the two population means differ.

ix. t Critical two-tail

This is the test’s crucial value. A t Critical value Calculator with 18 degrees of freedom and a 95% confidence level can be used to calculate this number.

In this instance, the critical value is 2.10092204 . We cannot reject the null hypothesis because our test statistic t is less than this number. Again, we lack adequate information to conclude that the two population means are distinct.

Things to Remember

• Excel demands that your data be arranged in columns, with data from each group in a separate column. The first row should have labels or headers.
• Clearly state your null hypothesis (usually that there is no significant difference between the group means) and your alternative hypothesis (the opposite of the null hypothesis).
• As a result of the t-test, Excel returns the p-value. A little p-value (usually less than the specified alpha level) indicates that the null hypothesis may be rejected and that there is a substantial difference between the group means.

Frequently Asked Questions

1. Can I perform a t-test on unequal sample sizes in Excel?

Answer: Yes, you can use the T.TEST function to do a t-test on unequal sample sizes. When calculating the test statistic, Excel automatically accounts for unequal sample sizes.

2. What is the difference between a one-tailed and a two-tailed t-test?

Answer: A one-tailed t-test determines if the means of the two groups differ substantially in a given direction (e.g., greater or smaller). A two-tailed t-test looks for any significant difference, regardless of direction.

3. Can I calculate effect size in Excel for t-tests?

Answer: While there is no built-in tool in Excel to calculate effect size, you may manually compute Cohen’s d for independent t-tests and paired sample correlations for paired t-tests using Excel’s basic mathematical operations.

In the end, we can conclude that you will learn some basic ideas on how to do a t Test in Excel. If you have any questions or feedback regarding this article, please share them in the comment section. Your valuable ideas will enrich my Excel expertise and hence the content of my upcoming articles.

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Excel Tutorial: How To Do A Hypothesis Test In Excel

Introduction.

Welcome to our Excel tutorial on how to conduct a hypothesis test using Excel. Hypothesis testing is a crucial component of statistical analysis, allowing us to make inferences about a population based on sample data. Using Excel for hypothesis testing offers several advantages, including its familiarity, ease of use, and the ability to perform complex statistical calculations with just a few clicks.

Key Takeaways

• Hypothesis testing is essential for making inferences about a population based on sample data.
• Using Excel for hypothesis testing offers familiarity, ease of use, and the ability to perform complex statistical calculations.
• Organizing and formatting data correctly in Excel is crucial for hypothesis testing.
• Understanding the different types of hypothesis tests and selecting the appropriate test is important for accurate analysis.
• Interpreting the results of the hypothesis test and avoiding common mistakes is essential for making valid conclusions.

Setting up the data in Excel

When conducting a hypothesis test in Excel, it is crucial to properly organize and format your data in a spreadsheet. This will ensure accurate and reliable results.

• Start by opening a new Excel spreadsheet and entering your raw data into the cells. It is important to have a clear understanding of the variables you are working with and how they relate to each other.
• Label each column with a clear and descriptive header to identify the variables being tested. This will help you keep track of the data and make it easier to analyze.
• Arrange the data in a logical and organized manner, such as grouping similar data together and using separate columns for different variables.
• Check that the data is formatted correctly, especially if it includes dates, currency, or percentages. Use the appropriate formatting options in Excel to ensure the data is displayed accurately.
• Remove any unnecessary formatting, such as extra spaces or special characters, to avoid errors in the analysis process.
• Double-check for any missing or erroneous data entries, and make sure that the data is complete and accurate before proceeding with the hypothesis test.

Choosing the Appropriate Test in Excel

When conducting a hypothesis test in Excel, it's crucial to choose the right test for your specific scenario. Understanding the different types of hypothesis tests and how to select the appropriate one is essential for accurate and meaningful results.

Parametric Tests:

Nonparametric tests:, one-sample, two-sample, and paired tests:, goodness-of-fit tests:, chi-square tests:.

Choosing the right hypothesis test in Excel requires careful consideration of the nature of the data and the specific research question. Here are some key factors to consider when selecting the appropriate test:

• Understanding the Data: Determine whether the data is continuous or categorical, and whether it follows a specific distribution.
• Research Question: Clearly define the research question and the type of comparison or relationship being investigated.
• Sample Size: Consider the size of the sample and whether it meets the assumptions of the chosen test.
• Dependent or Independent Variables: Determine whether the variables are independent or related in some way, as this will impact the choice of test.
• Assumptions: Ensure that the chosen test aligns with any specific assumptions or conditions required for accurate results.

Conducting the hypothesis test

When it comes to conducting a hypothesis test in Excel, there are a few key steps to follow in order to ensure accurate results. These steps include using the Data Analysis Toolpak and inputting the necessary parameters for the test.

The Data Analysis Toolpak is a powerful add-in for Excel that provides a variety of data analysis tools, including the ability to conduct hypothesis tests. To access the Toolpak, simply go to the "Data" tab, click on "Data Analysis" in the Analysis group, and select "t-Test: Two-Sample Assuming Equal Variances" for a two-sample t-test, or "t-Test: Paired Two Sample for Means" for a paired t-test.

Once the Data Analysis Toolpak is open, you will need to input the necessary parameters for the hypothesis test. This includes selecting the appropriate variables for analysis, specifying the significance level, and choosing whether to perform a one-tailed or two-tailed test. It is important to carefully review and input the correct parameters to ensure the accuracy of the test results.

By using the Data Analysis Toolpak in Excel and inputting the necessary parameters for the hypothesis test, you can effectively conduct hypothesis tests and analyze your data with confidence.

Interpreting the results

After performing a hypothesis test in Excel, it is important to understand how to interpret the results and make conclusions based on the data.

Identify the test statistic:

Look at the p-value:, consider the confidence interval:, check for statistical significance:, reject or fail to reject the null hypothesis:, consider the practical significance:, communicate the findings:, common mistakes to avoid.

When conducting a hypothesis test in Excel, there are some common mistakes that researchers often make. By being aware of these pitfalls, you can ensure that your results are accurate and reliable.

One of the most common mistakes when doing a hypothesis test in Excel is misinterpreting the results. It's important to carefully analyze the output of the test and understand what it is telling you. Avoid jumping to conclusions without thoroughly examining the data and the significance level.

Another mistake to avoid is using the wrong test for the hypothesis you are trying to test. Excel offers a variety of hypothesis tests, such as t-tests, F-tests, and chi-squared tests, among others. It's crucial to select the appropriate test for your specific research question and data set. Using the wrong test can lead to inaccurate results and conclusions.

In conclusion, hypothesis testing in Excel is a crucial tool for making data-driven decisions in various fields, from business to science. By using Excel, we can effectively analyze data and draw meaningful conclusions about our hypotheses.

As with any skill, practice makes perfect . So, I encourage you to continue exploring and practicing hypothesis testing in Excel. There are numerous resources available online that provide additional guidance and examples to help you master this valuable technique.

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The Complete Guide: Hypothesis Testing in Excel

In statistics, a hypothesis test is used to test some assumption about a population parameter .

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.

This tutorial explains how to perform the following types of hypothesis tests in Excel:

• One sample t-test
• Two sample t-test
• Paired samples t-test
• One proportion z-test
• Two proportion z-test

Let’s jump in!

Example 1: One Sample t-test in Excel

A one sample t-test is used to test whether or not the mean of a population is equal to some value.

For example, suppose a botanist wants to know if the mean height of a certain species of plant is equal to 15 inches.

To test this, she collects a random sample of 12 plants and records each of their heights in inches.

She would write the hypotheses for this particular one sample t-test as follows:

• H 0 :  µ = 15
• H A :  µ ≠15

Refer to this tutorial for a step-by-step explanation of how to perform this hypothesis test in Excel.

Example 2: Two Sample t-test in Excel

A two sample t-test is used to test whether or not the means of two populations are equal.

For example, suppose researchers want to know whether or not two different species of plants have the same mean height.

To test this, they collect a random sample of 20 plants from each species and measure their heights.

The researchers would write the hypotheses for this particular two sample t-test as follows:

• H 0 :  µ 1 = µ 2
• H A :  µ 1 ≠ µ 2

Example 3: Paired Samples t-test in Excel

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether a certain study program significantly impacts student performance on a particular exam.

To test this, we have 20 students in a class take a pre-test. Then, we have each of the students participate in the study program for two weeks. Then, the students retake a post-test of similar difficulty.

We would write the hypotheses for this particular two sample t-test as follows:

• H 0 :  µ pre = µ post
• H A :  µ pre ≠ µ post

Example 4: One Proportion z-test in Excel

A  one proportion z-test  is used to compare an observed proportion to a theoretical one.

For example, suppose a phone company claims that 90% of its customers are satisfied with their service.

To test this claim, an independent researcher gathered a simple random sample of 200 customers and asked them if they are satisfied with their service.

• H 0 : p = 0.90
• H A : p ≠ 0.90

Example 5: Two Proportion z-test in Excel

A two proportion z-test is used to test for a difference between two population proportions.

For example, suppose a s uperintendent of a school district claims that the percentage of students who prefer chocolate milk over regular milk in school cafeterias is the same for school 1 and school 2.

To test this claim, an independent researcher obtains a simple random sample of 100 students from each school and surveys them about their preferences.

• H 0 : p 1 = p 2
• H A : p 1  ≠ p 2

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Home » Statistical Analysis Excel » Hypothesis Testing

Struggling with Hypothesis Testing in Excel?

Qi macros makes hypothesis testing easy, even if you don't know anything about statistics.

Run Any Hypothesis Test using QI Macros

• Select your data.
• Click on QI Macros menu > Statistical Tools > the test you want
• QI Macros will do the math and analysis for you.

What is a Hypothesis Test?

A hypothesis test helps identify ways to reduce costs and improve quality. Hypothesis testing asks the question: Are two or more sets of data the same or different, statistically.

For companies working to improve operations, hypothesis tests help identify differences between machines, formulas, raw materials, etc. and whether the differences are statistically significant or not. Without such testing, teams can run around changing machine settings, formulas and so on causing more variation. These knee-jerk responses can amplify variation and cause more problems than doing nothing at all.

Three Types of Hypothesis Tests

• Classical Method - comparing a test statistic to a critical value
• p Value Method - the probability of a test statistic being contrary to the null hypothesis
• Confidence Interval Method - is the test statistic between or outside of the confidence interval

How to Conduct a Hypothesis Test

• Define the  null (H0) and an alternate (Ha) hypothesis .
• Conduct the test.
• Calculate the test statistic and the critical value (t-Test, F-test, z-Test, ANOVA, etc.).
• Calculate a p value and compare it to a significance level (a) or confidence level (1-a).
• Interpret the results to determine if you "cannot reject null hypothesis (accept null hypothesis)" or "reject the null hypothesis."

QI Macros for Excel Makes Hypothesis Testing as Easy as 1-2-3!

QI Macros adds a new tab to Excel's menu:

• Just input your data into an Excel spreadsheet and select it.
• Click on QI Macros menu , Statistical Tools and the test you want to run (t test, f test, z test, ANOVA, etc.).  If you are not sure which test to run, QI Macros Stat Wizard will analyze your data and run the possible tests for you.
• QI Macros performs all of the calculations AND interprets the results for you:

QI Macros Will Also Draw Charts to Help You Visualize the Differences in Your Data Sets

Cheat Sheet to Help You Interpret the Results Yourself

Stop struggling with hypothesis tests start conducting hypothesis tests in just minutes., download a free 30-day trial. run hypothesis tests now, qi macros can draw these charts too.

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2.1.3: The Research Hypothesis and the Null Hypothesis

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Hypotheses are predictions of expected findings.

The Research Hypothesis

A research hypothesis is a mathematical way of stating a research question. A research hypothesis names the groups (we'll start with a sample and a population), what was measured, and which we think will have a higher mean. The last one gives the research hypothesis a direction. In other words, a research hypothesis should include:

• The name of the groups being compared. This is sometimes considered the IV.
• What was measured. This is the DV.
• Which group are we predicting will have the higher mean.

There are two types of research hypotheses related to sample means and population means: Directional Research Hypotheses and Non-Directional Research Hypotheses

Directional Research Hypothesis

If we expect our obtained sample mean to be above or below the other group's mean (the population mean, for example), we have a directional hypothesis. There are two options:

• Symbol: \( \displaystyle \bar{X} > \mu \)
• (The mean of the sample is greater than than the mean of the population.)
• Symbol: \( \displaystyle \bar{X} < \mu \)
• (The mean of the sample is less than than mean of the population.)

Example \(\PageIndex{1}\)

A study by Blackwell, Trzesniewski, and Dweck (2007) measured growth mindset and how long the junior high student participants spent on their math homework. What’s a directional hypothesis for how scoring higher on growth mindset (compared to the population of junior high students) would be related to how long students spent on their homework? Write this out in words and symbols.

Answer in Words: Students who scored high on growth mindset would spend more time on their homework than the population of junior high students.

Answer in Symbols: \( \displaystyle \bar{X} > \mu \)

Non-Directional Research Hypothesis

A non-directional hypothesis states that the means will be different, but does not specify which will be higher. In reality, there is rarely a situation in which we actually don't want one group to be higher than the other, so we will focus on directional research hypotheses. There is only one option for a non-directional research hypothesis: "The sample mean differs from the population mean." These types of research hypotheses don’t give a direction, the hypothesis doesn’t say which will be higher or lower.

A non-directional research hypothesis in symbols should look like this: \( \displaystyle \bar{X} \neq \mu \) (The mean of the sample is not equal to the mean of the population).

Exercise \(\PageIndex{1}\)

What’s a non-directional hypothesis for how scoring higher on growth mindset higher on growth mindset (compared to the population of junior high students) would be related to how long students spent on their homework (Blackwell, Trzesniewski, & Dweck, 2007)? Write this out in words and symbols.

Answer in Words: Students who scored high on growth mindset would spend a different amount of time on their homework than the population of junior high students.

Answer in Symbols: \( \displaystyle \bar{X} \neq \mu \)

See how a non-directional research hypothesis doesn't really make sense? The big issue is not if the two groups differ, but if one group seems to improve what was measured (if having a growth mindset leads to more time spent on math homework). This textbook will only use directional research hypotheses because researchers almost always have a predicted direction (meaning that we almost always know which group we think will score higher).

The Null Hypothesis

The hypothesis that an apparent effect is due to chance is called the null hypothesis, written \(H_0\) (“H-naught”). We usually test this through comparing an experimental group to a comparison (control) group. This null hypothesis can be written as:

\[\mathrm{H}_{0}: \bar{X} = \mu \nonumber \]

For most of this textbook, the null hypothesis is that the means of the two groups are similar. Much later, the null hypothesis will be that there is no relationship between the two groups. Either way, remember that a null hypothesis is always saying that nothing is different.

This is where descriptive statistics diverge from inferential statistics. We know what the value of \(\overline{\mathrm{X}}\) is – it’s not a mystery or a question, it is what we observed from the sample. What we are using inferential statistics to do is infer whether this sample's descriptive statistics probably represents the population's descriptive statistics. This is the null hypothesis, that the two groups are similar.

Keep in mind that the null hypothesis is typically the opposite of the research hypothesis. A research hypothesis for the ESP example is that those in my sample who say that they have ESP would get more correct answers than the population would get correct, while the null hypothesis is that the average number correct for the two groups will be similar.

In general, the null hypothesis is the idea that nothing is going on: there is no effect of our treatment, no relation between our variables, and no difference in our sample mean from what we expected about the population mean. This is always our baseline starting assumption, and it is what we seek to reject. If we are trying to treat depression, we want to find a difference in average symptoms between our treatment and control groups. If we are trying to predict job performance, we want to find a relation between conscientiousness and evaluation scores. However, until we have evidence against it, we must use the null hypothesis as our starting point.

In sum, the null hypothesis is always : There is no difference between the groups’ means OR There is no relationship between the variables .

In the next chapter, the null hypothesis is that there’s no difference between the sample mean and population mean. In other words:

• There is no mean difference between the sample and population.
• The mean of the sample is the same as the mean of a specific population.
• \(\mathrm{H}_{0}: \bar{X} = \mu \nonumber \)
• We expect our sample’s mean to be same as the population mean.

Exercise \(\PageIndex{2}\)

A study by Blackwell, Trzesniewski, and Dweck (2007) measured growth mindset and how long the junior high student participants spent on their math homework. What’s the null hypothesis for scoring higher on growth mindset (compared to the population of junior high students) and how long students spent on their homework? Write this out in words and symbols.

Answer in Words: Students who scored high on growth mindset would spend a similar amount of time on their homework as the population of junior high students.

Answer in Symbols: \( \bar{X} = \mu \)

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Comparing Two Proportions

There is a statistical test – comparing two proportions – that helps us do this.  To use this test, data is collected from each of the two populations (for example, men and women).   The data collected is the sample size, the number of successes and the number of failures for each population.  These data can then be used to determine if there is a statistically significant difference in the results from the two populations.  This publication describes how to do this.

In this publication:

The Hypothesis

Interpretation of the math, pooled estimate of p, other considerations, quick links.

Please feel free to leave a comment at the end of the article.  You can also download a pdf copy of this publication at this link.

We will introduce the calculations using an example involving high school students and smoking.  One city, City A, has implemented a new smoking cessation program aimed at high school students.  Another city, City B, has not implemented the program.  We want to know if there is a statistically significant difference  in the two cities in the proportion of high school students who have smoked during the last six months.

In City A, 210 students from a sample of 950 high school students had smoked cigarettes in the last six months.  In City B, 255 students from a sample of 1300 high school students have smoked cigarettes in the last six months.  We want to know if there is a statistically significant difference between the two cities.

This situation represents where the two proportion test can be used.  You have a population of high school students in population 1 (City A).  And you have a population of high school students in population 2 (City B).  It doesn’t matter which one is population 1 or population 2.  For each population there is a “true” value for the proportion of high school students who have  not smoked cigarettes in the last six months.  We will denote those “true” proportions as p 1 and p 2 , respectively for population 1 and population 2.

You usually can’t ask each student in each city about smoking cigarettes to determine the “true” proportions.  Instead, you take a sample from each city. In this example, we took a sample of 950 from City A and a sample of 1300 from City B.  For each sample, certain statistics are calculated as shown in the table below.

Table 1: Two Proportions Calculations

Note that q ̂ 1   = 1 – p̂ 1 for City A.  The same holds for City B.

The proportion who has not smoked in the last 6 months is 0.779 for City A and 0.804 for City B.  It is not surprising that there is a difference between the two numbers.  After all, we  are taking a sample and there is variation present.  The question is whether or not the two numbers are significantly different statistically.  The two proportion test will answer that question for us.

The two proportion test described below is the large sample case.  The large case is used if the following is true:

We will use the two-tailed test in this example.  The null hypothesis (H 0 ) for the two-tailed test is that the two population proportions are the same:

H 0 :  p 1 – p 2 = 0

The alternate hypothesis (H 1 ) is that the two population parameters are not equal:

H 1 :  p 1 – p 2 ≠ 0

For large samples, you can assume that the samples from the populations are normally distributed.  This allows you to calculate the z statistic to help determine if the difference between the two proportions is zero or not.  The z statistic measures how many standard deviations a value is from the hypothesized difference of 0.

There are two ways to tell if there is a significant difference between the two proportions.  One looks at the 95% confidence interval around p 1 – p 2 .  If this interval contains 0, we will conclude that there is no evidence that the proportions are different (e.g., accept H 0 ).  If the 95% confidence limit does not contain zero, we will conclude that there is evidence that the proportions are not equal. 

The second method looks at the calculated p-value.  The p-value can be looked at as the probability of getting the calculated z statistic if the H 0 is true.  If that probability is small, then we conclude that there is evidence that the proportions are not equal.  “Small” is determined by the value of alpha you select.  Usually, alpha is 0.05.   The confidence limits are set at 1 – alpha, or 1 – .05 = 95% when alpha = 0.05.

We will start by calculating the z statistic.  We need a measure of variation to do that.  The variance is estimated as the following:

The 100(1- s )% two-sided confidence interval for p 1 – p 2 is then:

The z s /2 in the equation above is not the z statistic.  The z term, z s /2 , is referring to the z critical value from a z table that corresponds to s /2.   Since this is a two-sided test, you divide alpha by two.  In this example, s /2 = 0.025 since we chose alpha = 0.05.  The value of  z s /2 for alpha = 0.05 is 1.96.

Let’s begin to put some numbers in to do the calculations, starting with the variance:

Now, calculate the z value:

Finally, calculate the 95% confidence interval:

We have done the calculations; now to interpret the results.  As stated before, there are two ways we will determine if the two proportions are the same or different.  One of those is the 95% confidence interval.  The lower confidence limit for this example is -0.059; the upper confidence interval is 0.009.  All values between the confidence limits are possible values for the difference in the two proportions.  Since the confidence interval contains 0, we conclude that is possible for the two proportions to be equal and accept the null hypothesis.

You can also see this easily in the plot below.  The hypothesized difference is within the confidence limits.  There is no evidence that the two cities have a different proportion of high school students who have not smoked in the last six months. 

The second method of determining if there is a difference in the two proportions is through the z value.  The z statistic for this example is -1.44.   The question to answer is what is the probability of getting this z statistic if the null hypothesis is true – that there is no difference in the two proportions.  To find this probability, you can use the following function in Excel (for the two-sided confidence interval):

p-value =2*(1-NORM.S.DIST(ABS(z),TRUE)

This gives a p-value of 0.15.  This means that there is about a 15% probability of getting this z value or one more extreme if the null hypothesis is true.  We chose alpha = 0.05.  Since the p-value is greater than alpha, we conclude that there is no evidence that the two proportions are not equal.  The null hypothesis is accepted.

It should be noted that sometimes the two p values are pooled to get a “better” estimate of the true proportion.  If H 0 is true, then both p 1 and p 2 are estimating the same true population, p.  The pooled estimate of p is given by:

The  pooled estimate for this example is 0.793.  The z statistic equation is slightly different and is then given by:

The variance of the pooled estimate is given by:

The 95% confidence limits are given by:

The z statistic and the  95% confidence limits are essentially the same as before.  This is because the two proportions are essentially the same.  You will see differences when there are larger differences in the two proportions.

The following example is from Penn State’s online statistical course. 

“Time magazine reported the result of a telephone poll of 800 adult Americans. The question posed of the Americans who were surveyed was: “Should the federal tax on cigarettes be raised to pay for health care reform?” The results of the survey were:”

These data were analyzed using the SPC for Excel software .  The output is shown below.

The output shows the results of the calculations covered above.  The p-value is very small (essentially zero) and the confidence interval does not include 0, so there is evidence that the difference between smokers and non-smokers is not 0 and the null hypothesis is rejected. The chart below (part of the output from the SPC for Excel software) confirms this.

The hypothesized difference in proportions is outside the confidence limits.

The procedure outlined in this publication is the large sample case.  There is also a procedure for the small sample case.  This procedure will not be covered here but it is done in the SPC for Excel software as well.  It is called the Fisher’s Exact Test.  The p-value for that test is given in the output shown above.  You can also perform one-sided confidence intervals and not just the two-sided process shown here.

This publication showed the mathematics behind comparing two proportions to determine if they are equal or not.  The two things to help decide that are the confidence interval and the probability of getting the calculated z statistic.   If the confidence interval does not contain 0, there is evidence that the two proportions are not the same.  If the p-value is small, there is also evidence that the two proportions are not the same.

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Thanks so much for reading our SPC Knowledge Base. We hope you find it informative and useful. Happy charting and may the data always support your position.

Dr. Bill McNeese BPI Consulting, LLC

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How to Perform the Friedman Test in R

The Friedman Test is a non-parametric statistical test used to determine if there are statistically significant differences between multiple related groups. It is often employed when the outcome variable is ordinal or continuous and the independent variable is categorical with three or more levels. This comprehensive guide will walk you through the process of conducting the Friedman Test in R, including data preparation, hypothesis testing, and result interpretation.

Here are the step-by-step explanations of the Friedman Test in R Programming Language .

Step 1. Install and Load Required Packages

Before conducting the Friedman Test, make sure you have the necessary packages installed and loaded:

Step 2. Prepare Your Data

Start by loading your dataset into R and ensuring that it is structured appropriately for analysis. The dataset should contain one or more outcome variables (dependent variables) and a categorical factor variable (independent variable) with three or more levels.

Step 3. Conduct the Friedman Test

Use the friedman.test() function from the PMCMRplus package to conduct the Friedman Test:

Replace outcome with the name of your outcome variable, group with the name of your categorical factor variable, and your_data with the name of your dataset.

Step 4. Interpret the Results

After conducting the Friedman Test, you can obtain the test statistics, degrees of freedom, and p-value:

If the p-value is less than your chosen significance level (e.g., 0.05), you can reject the null hypothesis and conclude that there are statistically significant differences between the groups. Otherwise, you fail to reject the null hypothesis.

Conducting the Friedman Test in R

Here’s a complete example demonstrating how to conduct the Friedman Test in R using a hypothetical dataset:

The p-value is (4.575e-08), we reject the null hypothesis. This means there is strong evidence to conclude that there are significant differences in the ranks of the outcome variable among the groups A, B, and C.

The Friedman rank sum test results indicate that the differences in ranks among the groups are statistically significant, as evidenced by the high test statistic and the extremely low p-value. This suggests that at least one of the groups is different from the others in terms of the outcome variable. The box plot visualization helps to see these differences more clearly.

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