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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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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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Hypothesis Testing

So, a hypothesis is just a statement of theory.  It may or may not be true.  A drug company can claim that a new drug is better at decreasing blood pressure.   You may claim that the diet plan you created helps people lose more weight than a nationally known diet plan.  All these things are just statements – just hypotheses.

The hypothesis is the starting point.  From there, we have to test the hypothesis and reach a decision if the hypothesis is probably true or probably false.  Note the word “probably.”  There is always variation – so there is always a chance for you to make the wrong decision.  This month’s publication takes a look at the five steps involved in conducting a hypothesis test.

In this issue:

• The problem
• A brief pause for the standard normal distribution
• Formulate the null hypothesis and the alternative hypothesis
• Determine the significance level
• Collect the data and calculate the sample statistics
• Calculate the p value for the hypothesis test
• Compare the p value to the desired significance level

Quick Links

You can download this publication as a pdf here .

The Problem

The average coating thickness is 5 mil.  You want to be sure that the coating thickness remains the same before you will approve the process change.

The team wants to perform a hypothesis test to prove that the average coating thickness will not change.  The team will go through the basic five steps of hypothesis testing:

The details of the five steps are shown below.  However, before those steps are covered, a review of the standard normal distribution is needed.  This will be required when we do some calculations.

A Brief Pause for the Standard Normal Distribution

We need to digress a moment here because we will need to make use of a special case of the normal distribution – when the average = 0 and the standard deviation = 1. This special case is called the standard normal distribution and is shown in Figure 1.

Figure 1: Standard Normal Distribution

For this distribution, the area under the curve from -∞ to +∞ is equal to 1.0. In addition, the area under the curve is proportional to the fraction of measurements that fall in that region. These two facts can used to help determine the fraction of measurements that fall above some value (such as a specification limit), below some value, or between two values.

z=  (x- μ)/σ

where x is some value, μ is the average, and σ is the standard deviation of the x values.  The value of z (the z score) is simply how many standard deviations a value, x, is from the average.

For example, suppose x is 1.5 standard deviations below the average.  In this case, z = -1.5.  The area below z = -1.5 is the percentage of x values that are more than 1.5 standard deviations below the average.  For z = -1.5, that area is 6.68% as is shown in Figure 1.   If z = 1.5, then the area above z = 1.5 is the percentage of x values that are more than 1.5 standard deviations above the average.  This area is also 6.68%.

To find the percentage of data within z = -1.5 and z = 1.5, you simply use the fact that the area under the curve is 100%, so the percentage of data between the two z values is 100 – 6.68 – 6.68 = 86.64%.  You can determine these percentages from a table of z values (see our publication on the normal distribution ) or by using Excel’s NORMSDIST function.

These percentages can also be viewed as probabilities, e.g., the probability of getting a result that is less than -1.5 standard deviations below the average is 0.0668.  We will make use of this knowledge below.  Now back to the steps in hypothesis testing.

Step 1: Formulate the Null Hypothesis and Alternative Hypothesis

So the null hypothesis (H 0 ) is that the process change will not impact the average coating thickness; the average coating thickness (μ) will remain at 5.  This is usually written as:

Now for the alternative hypothesis, which is denoted by H 1 .  The alternative hypothesis is that the process change will have an effect on the average coating thickness and the average coating thickness will not equal 5.  This is usually written as:

This is called a two-sided hypothesis test since you are only interested if the mean is not equal to 5.  You can have one-sided tests where you want the mean to be greater than or less than some value.

Step 2: Determine the Significance Level You Want

The significance level is important in hypothesis testing.  It is the probability of rejecting the null hypothesis when it is true.  This probability is denoted by α.  Typical values of  α include 0.05 and 0.01.  You decide that you want α to be 0.05.  This means that there is only a 5% of chance of rejecting the null hypothesis when it is actually true.

Step 3: Collect the Data and Calculate the Sample Statistics

X   = average coating thickness = 5.06

s = standard deviation of the coating thickness = 0.20

We have our statistics.  How do you decide to accept or reject the null hypothesis?  The way you do this is to assume that the null hypothesis is true and then determine the probability (p value) of getting this sample average.  If the p value is large, it means that there is large probability of getting an average thickness of 5.06 with a standard deviation of 0.20 when the null hypothesis is true and you will accept that the null hypothesis is probably true.  But if the probability of getting these statistics is small, you will assume that the null hypothesis is probably not true and reject it in favor the alternative hypothesis.

Step 4: Calculate the p Value

To determine this probability, you will need to consider your sampling distribution.    The distribution of sample averages tends to be normal when the sample size is large enough.  We will use this assumption here.  So, your sampling distribution is represented by all the possible sample averages of sample size 25 from the population of coating thicknesses.  This normal distribution is shown in Figure 2.

Figure 2: Normal Distribution for Sample Averages

The highest point on the curve is the average.  The population average of the sample averages (μ X ) is equal to the population average, μ, so we have just used μ in Figure 1.  The standard deviation of the sample averages is denoted by σ X .

To be able to draw your sampling distribution, you need to know μ X   and  σ X .  Since you assumed that the null hypothesis is true, μ X   = 5.0.  The standard deviation of the sample averages is given by:

where σ is the population standard deviation and n is the sample size.

You don’t know what the population standard deviation is, but you have an estimate from the sample statistics.  The standard deviation of the 25 samples was 0.2.  You can use this as the population standard deviation.

σ X =σ/√n =  s/√n=0.2/√25=0.04

Now you can draw the sampling distribution and add the sample average as shown in Figure 3.

Figure 3: Sampling Distribution

Now we return to the z score.  Remember, the z score is a measure of how many standard deviations the sample average ( X  )is from the population average (μ).   For this example, the z value is calculated as:

z=  ( X -μ)/σ X =(5.06-5)/.04=.06/.04=1.5

So, 5.06 is 1.5 standard deviations away from the average.    As shown above, the probability of getting a result that is 1.5 standard deviations away from the average is 0.0668.  Remember, this a two-side test, so you didn’t care if the difference was above or below the average.  So, the probability of getting an average that is more than 1.5 standard deviations away from the average is 2(0.0668) = 0.1336 or 13.36%.  This is the p value:

p value = 0.1336

Remember what the p value represents.  You assumed that the null hypothesis is true.  The p value is the probability of getting this result (or a more extreme result) if the null hypothesis is true.

Step 5:  Compare the p value to the Desired Significance Level

In step 2, we set the significance level at 0.05.  Since our p value is greater than this, we conclude that the coating thickness was not impacted by the process change.  We accept the null hypothesis as probably being true.  If the p value had been less than 0.05, we would rejected the null hypothesis and said that the process change did impact the coating thickness.

This newsletter has taken a look at how to perform hypothesis testing.  The five steps are:

• Determine the significance level you want

The normal distribution was used to demonstrate how hypothesis testing is done.  You will not always be dealing with the normal distribution but the process is essentially the same.  One item that is still to be discussed is how to select the sample size.  This will be the subject of a later publication.

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#1 Excel tutorial on the net

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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Statistics for Beginners in Excel – Null and Alternative Hypothesis

(basic statistics for citizen data scientist), null and alternative hypothesis.

Generally to understand some characteristic of the general population we take a random sample and study the corresponding property of the sample. We then determine whether any conclusions we reach about the sample are representative of the population.

This is done by choosing an  estimator  function for the characteristic (of the population) we want to study and then applying this function to the sample to obtain an  estimate . By using the appropriate statistical test we then determine whether this estimate is based solely on chance.

The hypothesis that the estimate is based solely on chance is called the  null hypothesis . Thus, the null hypothesis is true if the observed data (in the sample) do not differ from what would be expected on the basis of chance alone. The complement of the null hypothesis is called the  alternative hypothesis .

The null hypothesis is typically abbreviated as H 0  and the alternative hypothesis as H 1 . Since the two are complementary (i.e. H 0  is true if and only if H 1  is false), it is sufficient to define the null hypothesis.

Since our sample usually only contains a subset of the data in the population, we cannot be absolutely certain as to whether the null hypothesis is true or not. We can merely gather information (via statistical tests) to determine whether it is likely or not. We therefore speak about  rejecting  or  not rejecting  (aka  retaining ) the null hypothesis on the basis of some test, but not of  accepting  the null hypothesis or the alternative hypothesis. Often in an experiment we are actually testing the validity of the alternative hypothesis by testing whether to reject the null hypothesis.

When performing such tests, there is some chance that we will reach the wrong conclusion. There are two types of  errors :

• Type I  – H 0  is rejected even though it is true ( false positive )
• Type II  – H 0  is not rejected even though it is false ( false negative )

The acceptable level of a Type I error is designated by  alpha  ( α ), while the acceptable level of a Type II error is designated  beta  ( β ).

We use the following terminology:

Significance level  is the acceptable level of type I error, denoted  α . Typically, a significance level of  α  = .05 is used (although sometimes other levels such as  α  = .01 may be employed). This means that we are willing to tolerate up to 5% of type I errors, i.e. we are willing to accept the fact that in 1 out of every 20 samples we reject the null hypothesis even though it is true.

P-value  (the  probability value ) is the value  p  of the statistic used to test the null hypothesis. If   p  <  α  then we reject the null hypothesis.

Critical region  is the part of the sample space that corresponds to the rejection of the null hypothesis, i.e. the set of possible values of the test statistic which are better explained by the alternative hypothesis. The significance level is the probability that the test statistic will fall within the critical region when the null hypothesis is assumed.

Usually the critical region is depicted as a region under a curve for continuous distributions (or a portion of a bar chart for discrete distributions).

The typical approach for testing a null hypothesis is to select a statistic based on a sample of fixed size, calculate the value of the statistic for the sample and then reject the null hypothesis if and only if the statistic falls in the critical region.

One-tailed hypothesis testing  specifies a direction of the statistical test. For example to test whether cloud seeding increases the average annual rainfall in an area which usually has an average annual rainfall of 20 cm, we define the null and alternative hypotheses as follows, where  μ  represents the average rainfall after cloud seeding.

H 0 :  µ  ≤ 20 (i.e. average rainfall does not increase after cloud seeding)

H 1 :  µ  > 20 (i.e. average rainfall increases after cloud seeding

Here the experimenters are quite sure that the cloud seeding will not significantly reduce rainfall, and so a one-tailed test is used where the critical region is as in the shaded area in Figure 1. The null hypothesis is rejected only if the test statistic falls in the critical region, i.e. the test statistic has a value larger than the critical value.

Figure 1 – Critical region is the right tail

The critical value here is the  right  (or  upper )  tail . It is quite possible to have one sided tests where the critical value is the  left  (or  lower )  tail . For example, suppose the cloud seeding is expected to decrease rainfall. Then the null hypothesis could be as follows:

H 0 :  µ  ≥ 20 (i.e. average rainfall does not decrease after cloud seeding)

H 1 :  µ  < 20 (i.e. average rain decreases after cloud seeding)

Figure 2 – Critical region is the left tail

Two-tailed hypothesis testing  doesn’t specify a direction of the test. For the cloud seeding example, it is more common to use a two-tailed test. Here the null and alternative hypotheses are as follows.

H 0 :  µ  = 20

H 1 :  µ  ≠ 20

The reasons for using a two-tailed test is that even though the experimenters expect cloud seeding to increase rainfall, it is possible that the reverse occurs and, in fact, a significant decrease in rainfall results. To take care of this possibility, a two tailed test is used with the critical region consisting of both the upper and lower tails.

Figure 3 – Two-tailed hypothesis testing

In this case we reject the null hypothesis if the test statistic falls in either side of the critical region. To achieve a significance level of  α , the critical region in each tail must have size  α /2.

Statistical power  is 1 –  β . Thus power is the probability that you find an effect when one exists, i.e. the probability of correctly rejecting a false null hypothesis. While a significance level for type I error of  α  = .05 is typically used, generally the target for  β  is .20 or .10, and so .80 or .90 is used as the target value for power.

The  general procedure  for null hypothesis testing is as follows:

• State the null and alternative hypotheses
• Specify  α  and the sample size
• Select an appropriate statistical test
• Collect data (note that the previous steps should be done prior to collecting data)
• Compute the test statistic based on the sample data
• Determine the p-value associated with the statistic
• Decide whether to reject the null hypothesis by comparing the p-value to  α  (i.e. reject the null hypothesis if  p < α )
• Report your results, including effect sizes (as described in Effect Size)

Observation : Suppose we perform a statistical test of the null hypothesis with  α  = .05 and obtain a p-value of  p  = .04, thereby rejecting the null hypothesis. This does not mean that there is a 4% probability of the null hypothesis being true, i.e.  P (H 0 ) =.04. What we have shown instead is that assuming the null hypothesis is true, the conditional probability that the sample data exhibits the obtained test statistic is 0.04; i.e.  P ( D |H 0 ) =.04 where  D  = the event that the sample data exhibits the observed test statistic.

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How to Find Test Statistic in Excel: A Step-by-Step Guide

In the realm of data analysis, a hypothesis test is a cornerstone. It’s the statistical equivalent of being a detective, piecing together whether the evidence (our data) supports our initial hunch about a population. And where do many of us turn to crunch those numbers? Excel, of course! It’s the trusty sidekick for many data analysts, researchers, and students alike.

Using Excel for a t-test, we can determine if two sets of data—one representing each population—are different from each other in a meaningful way. But it’s not just a matter of punching in numbers and hoping for the best. We need to understand which test to apply and interpret the results correctly. It’s the difference between finding a ‘smoking gun’ or just smoke and mirrors.

We tackle Excel’s powerhouse features, namely the Data Analysis Toolpak and the T.TEST function, to calculate our test statistics. Through these, Excel whisks through calculations, giving us the p-value and test statistic—a one-two punch to help us decide if our hypothesis holds water. If numbers could talk, these tests would be yelling out the answers to us. But fear not, we’ll keep the heavy math talk at bay and focus on clear, actionable steps.

• 1.1 Understanding Null and Alternative Hypotheses
• 1.2 Significance Level and Its Importance
• 2.1 Preparing Data for T-Test Analysis
• 2.2 Utilizing Excel Functions for T-Tests
• 2.3 Interpreting T-Test Results
• 3.1 Distinguishing Between Equal and Unequal Variances
• 4.1 Checking Assumptions Before Testing
• 4.2 Addressing Mistakes in Data Interpretation

Exploring Hypothesis Testing Fundamentals

Before we jump into the nitty-gritty of Excel and those test statistics, let us pave the ground with the bedrock concepts of hypothesis testing. It’s like building a house—you need a solid foundation, right? So let’s lay those bricks down.

Understanding Null and Alternative Hypotheses

Imagine you’re testing whether a new teaching method is more effective than the standard one. Our null hypothesis ( H 0 ) suggests that there’s no difference in effectiveness. The alternative hypothesis ( H A ), on the other side of the coin, indicates there is a difference.

Significance Level and Its Importance

When we talk about the significance level, often symbolized by alpha (α), think of it as the tightrope we walk on making decisions. Set too high, we risk falling into the pit of false discoveries. Too low, and we might not give our alternative hypothesis ( H A ) a fair chance to show its true colors.

Choosing the right α is critical, as it influences our p-value cut-off point for determining whether to reject H 0 . P-values below our chosen α suggest we might just be on to something. They wave a little flag that maybe, just maybe, our H A isn’t just a shot in the dark.

Performing T-Tests in Excel

Excel’s suite of tools, including the T.TEST function and the Analysis Toolpak, makes conducting statistical tests a walk in the park. We’re going to break down how to set up your data, use Excel’s functionality to run T-Tests, and then interpret the results with confidence.

Preparing Data for T-Test Analysis

Firstly, we need to ensure our data is T-Test ready. Properly segmented into two columns —each representing a sample or group — this is critical. Then we confirm that the data meets T-Test assumptions; the values should be independent and randomly selected from normal distributions.

Utilizing Excel Functions for T-Tests

Assuming the Analysis Toolpak is already enabled (you can find it in the ‘Add-ins’ section), conducting a T-Test is a breeze. Simply click on the ‘Data’ tab, look for the ‘Data Analysis’ button, and then select the appropriate T-Test type. Excel lays out clear input fields for your sample data, means, variances, and other necessary values.

Interpreting T-Test Results

Now, for the grand reveal: interpreting the results! After running the T-Test, you’ll get a test statistic and a p-value. Our test statistic reflects the difference between our sample means, normalized by the variability of our data, telling us the magnitude of the effect. The p-value, on the other hand, gives us the probability of seeing such results if the null hypothesis (no difference) is true.

Understanding Variance in T-Tests

In the world of Excel and t-tests, variance is like the secret sauce that can drastically change the flavor of your results. So, let’s roll up our sleeves and get the down-low on this spicy little number.

Distinguishing Between Equal and Unequal Variances

When we play the data detective with t-tests, we’re often confronted with a choice: are the variances from our two groups wearing the same-sized shoes (equal variances) or does one prefer a snug fit while the other likes room to wiggle (unequal variances)? This isn’t just academic navel-gazing—it’s central to choosing the right test and interpreting our data’s story accurately.

Equal Variances: The Classic Assumption

Imagine two groups squaring off—maybe brand A and B batteries. If we assume they have equal variance, we’re saying the consistency, that spicy little thing called standard deviation, is pretty much the same for both brands. We’d use the t-Test under the assumption of equal variances for this scenario. But remember, assuming can make a data analyst out of you and me—so we’ve got to check those variances with something like Levene’s Test first.

Unequal Variances: When Things Get Wobbly

Now let’s say brand A’s battery life is consistent as grandma’s secret sauce, while brand B is all over the place. We’ve got unequal variances on our hands, folks. Excel whispers to us, “psst, use the t-Test for unequal variances.” It’s like choosing different-sized boxing gloves for a fair fight—tailoring our approach to the data’s quirks.

Choosing between these Excel t-tests isn’t just academic; it’s a critical step on our journey to trustworthy results. Armed with the right test, we can strut into the data party confident that we’re interpreting the whispers and shouts of our datasets like the savvy statistical mavens we are.

Avoiding Common T-Test Pitfalls

When we dive into the world of t-tests in Excel, it’s like navigating a maze; thrilling, but full of possible wrong turns. Let’s guide you through some tricks of the trade to ensure you don’t hit a dead end.

Checking Assumptions Before Testing

Before you crunch the numbers, remember that t-tests have a VIP list of assumptions you must tick off. If your data sets have vastly different variances or aren’t normally distributed, Excel will churn out results that are less reliable than a chocolate teapot. So, ensure your data is homogenous like good old porridge and not too skewed – no one likes biased porridge.

Addressing Mistakes in Data Interpretation

Picture this: you’ve run the t-test, got the p-value, and you’re ready to present your findings. But slow down, partner! Don’t let the excitement trip you up at the finish line. A low p-value might seem like a green light, but if you’ve ignored effect size, you could be celebrating a victory lap in an empty stadium. Size matters – and in this case, it’s the effect size that tells us if our findings are practical ninjas or just statistical confetti.

• Misreading p-values – don’t confuse statistical with practical significance.
• Overlooking effect size – it’s the muscle behind the p-value punch.
• Glossing over assumptions – they’re the foundation, not an afterthought.

Remember, we’re in this together. By staying sharp and dodging these pitfalls, we keep our research robust and our conclusions valid. It’s the difference between being a statistical wizard and just waving a wand in Excel. Let’s crunch the numbers with our eyes wide open!

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Statistics Made Easy

How to Conduct a 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.

This tutorial explains how to conduct a one sample t-test in Excel.

Suppose a botanist wants to know if the mean height of a certain species of plant is equal to 15 inches. She collects a random sample of 12 plants and records each of their heights in inches.

The following image shows the height (in inches) for each plant in the sample:

We can use the following steps to conduct a one sample t-test to determine if the mean height for this species of plant is actually equal to 15 inches.

Step 1: Find the sample size, sample mean, and sample standard deviation.

First, we need to find the sample size, sample mean, and sample standard deviation, which will all be used to conduct the one sample t-test.

The following image shows the formulas we can use to calculate these values:

Step 2: Calculate the test statistic  t .

Next, we will calculate the test statistic  t  using the following formula:

t  = x – µ / (s/√ n )

x  = sample mean

µ = hypothesized population mean

s = sample standard deviation

n = sample size

The following image shows how to calculate  t  in Excel:

The test statistic  t  turns out to be  -1.68485 .

Step 3: Calculate the p-value of the test statistic.

Next, we need to calculate the p-value associated with the test statistic using the following function in Excel:

=T.DIST.2T(ABS(x), deg_freedom)

x = test statistic  t

deg_freedom = degrees of freedom for the test, which is calculated as n-1

Technical Notes:    The function T.DIST.2T() returns the p-value for a two-tailed t-test. If you’re instead conducting a left-tailed t-test or a right-tailed t-test, you would instead use the functions T.DIST() or T.DIST.RT() , respectively.

The following image shows how to calculate the p-value for our test statistic:

The p-value turns out to be  0.120145 .

Step 4: Interpret the results.

The two hypotheses for this particular one sample t test are as follows:

H 0 :  µ = 15 (the mean height for this species of plant is 15 inches)

H A :  µ ≠15 (the mean height is  not  15 inches)

Because the p-value of our test (0.120145)  is greater than alpha = 0.05, we fail to reject the null hypothesis of the test.

We do not have sufficient evidence to say that the mean height for this particular species of plant is different from 15 inches.

Additional Resources

The following tutorials explain how to perform other common types of t-tests in Excel:

How to Conduct a Two Sample t-Test in Excel How to Conduct a Paired Samples t-Test in Excel

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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Dartmouth Sports Analytics

The Designated Hitter Rule: Was the National League Right?

By Ava Politis '26

Introduction

Professional baseball's National League was the only  league to exist  until the American League was founded in 1901 ("American League is Founded"). While both teams were considered major leagues, the American League utilized the Designated Hitter rule introduced in 1973 ("What are the Differences Between the American and National Leagues?"). The designated hitter is a player who does not play defense and bats in place of the pitcher since pitchers are not known as the strongest hitters on the team ("Designated Hitter Rule | Glossary"). For example, from 1929-1941, pitcher Wes Ferrell only hit 38 home runs in his entire professional career, which is the most a pitcher has ever hit. On the other hand, David Ortiz hit 541 home runs as a designated hitter from 1997 to 2016 as he crushed 30-plus homers in 10 different seasons (" All time  home run leaders at each position"). When interleague play occurred, and an American League team was playing at a National League stadium, the pitcher would have to hit and vice versa. In 2022, the National League officially gave in and  officially  adopted the Designated Hitter rule. 

Although adding  a designated hitter to the National League would eliminate moments like Bartolo Colon hitting his famous home run in 2016, it could allow the National League to bring up more power hitters to the plate and rest their pitchers.  On the surface, having designated hitters instead of pitchers would create a more effective offense as they hit for power. They typically have more hits and home runs than an average pitcher, which would allow more runs to be produced per game. According to Braves pitcher Charlie Morton, "'I don't think anyone wants to watch me strike out on three pitches and walk back to the dugout 98 times out of a hundred '"  (Kurkjian).  With that being  said, some pitchers were also great hitters, as Madison Bumgarner hit two grand slams in his career, which is as many as Pete Rose and Derek Jeter had in 25,248 at-bats combined (Kurkjian). This paper will investigate whether or not there is statistically significant evidence supporting the hypothesis that designated hitters' offensive production in the National League is more effective than National League pitchers.

To examine whether or not designated hitters are more effective hitters than pitchers in the National League, I will examine National League designated hitters’ production from 2022 to 2023 and compare it to the offensive  production  of National League pitchers from 2002 to 2021. I will exclude data from 2020 due to the Covid-19 pandemic. To further dissect these two data sets, I will conduct hypothesis tests on runs batted in (RBI) per plate appearance and strikeout rate as my two parameters. Thus, the null hypothesis is that the runs batted in (RBI) per plate appearance and strikeout rate are the same regardless of whether the batter is a pitcher or a designated hitter in the National League.  The alternative hypothesis that I propose  is that the   strikeout rate decreases and the RBIs per plate appearance increase when the designated hitter is batting instead of a pitcher in the National League.  For strikeout rate, a one-tail t-test will be conducted as strikeout rate is the proportion of a result of a strikeout per plate appearance. A one-tail t-test can also be used for RBI, as it can be viewed as the number of runs batted  in  per plate appearance. On-base plus slugging (OPS) would be a possible statistic to review. However, because its constituent parts, on-base  percentage and slugging percentage , involve different denominators, namely plate appearances (PA) and at-bats (AB), it would be challenging to evaluate. However, if I were to compare the RBIs and strikeout rate of a National League designated hitter versus a pitcher, I would better understand how the designated hitter affects the offense. The strikeout rate will reveal which position is striking out more, leading to less productive at-bats. RBIs will reveal which position contributes more runs to a game, which is necessary for an effective offense and a win. For National League designated hitters, data from the past two MLB regular seasons will be used since that is when they were allowed to bat. For National League pitchers, data will be used from 2002 to 2021 MLB seasons, as 2002 is the farthest I can go back and when pitchers primarily batted. All data are collected from FanGraphs. 

Below are National League’s designated hitters’ counting statistics from the 2022 and 2023 MLB regular season.

The counting statistics were then utilized to calculate the total strikeout rate (K%) and RBIs per plate appearance.

The same process was used to calculate National League’s pitchers counting statistics and rates from the 2002 to 2021 MLB seasons (I am excluding 2020 due to the COVID-19 pandemic).

The data reveals that when a designated hitter replaces a pitcher in the line-up in the National League, the strikeout rate of the number nine spot decreases  and  the number of RBIs per plate appearance increases. Since the data matches my alternative hypotheses, I will conduct hypothesis tests to identify if the results are statistically significant.

To begin, I will  first analyze if the change in the batter affects the strikeout rate. During that period, National League designated hitters had a total of 11799 plate appearances and struck out 2762 times (Table 1). However, National league pitchers had a total of 50753 plate appearances and struck-out 17597 times (Table 3). Due to the calculations, pitchers had an average strikeout rate of 34.67% (Table 4) while the designated hitters struckout 23.41% of the time (Table 2).  

Knowing this information, I can conduct a t-test.  0.3467 is the sample mean  and the hypothesized mean is 0.2341.  The standard deviation of the sample is 0.15  as it was  calculated in Excel with a sample size of 19 since  there were  19 different seasons examined.  

Since the degree of freedom is 18, I find on the t-distribution chart 3.27 falls between levels 0.005 and 0.001 to get a value of 0.003 since I am doing a one-tailed test. Since the p-value is less than 0.01, I can reject the null hypothesis and accept the alternative hypothesis that strikeout rate decreases when the designated hitter  is batting  instead of a pitcher in the National League.

To go deeper into my  analysis, I will analyze if the change of the batter in the National League has an effect on the number of RBIs per plate appearance. National League designated hitters had a total of 11799 plate appearances with a total of 1525 RBIs (Table 1). However, National League pitchers had a total of 50753 plate appearances with 2468 RBIs (Table 3). Due to the calculations, pitchers averaged 0.05 RBIs per plate appearance (Table 4), while the designated hitters averaged 0.13 RBIs per plate appearance (Table 2).  

The sample mean is 0.05  and 0.13 is  the hypothesized mean.  With a standard deviation of 0.10 (calculated by Excel) and a sample size of 19, I could conduct a t-test.

The hypothesis testing implied that my results were statistically significant, meaning that replacing the designated hitter in the number nine slot in the lineup instead of the pitcher provided a more effective offense in the National League. In more depth, National League designated hitters  provide  a more productive at-bat since they strike out less and provide more runs for their team as they tend to hit with men on base. Even though strikeout rate and RBIs per PA  provide  a good representation of a good hitter, there are other ways of analyzing if a batter is effective. Analyzing statistics such as on-base percentage or slugging would have been  a good  way to represent if a batter can get on base and hit for power.  However, I wanted to see if designated hitters provide a more productive at-bat than pitchers, so it was best , in my opinion,  to analyze strikeout rate and RBIs.   Throughout my research,  it seems that  designated hitters provide a better offense than pitchers in the National League, and there  could be  multiple reasons for this.   Pitchers  do not  put  a lot of  focus into batting as they need to focus on pitching.  Starting pitchers have pressure on them to go far into the game and give up the least amount of runs possible.  Hitting a home run is  most likely  the last thing on their mind.  Designated hitters only have to focus on hitting. They do not have to practice defensive skills and mostly have to worry about offense, giving them more time to improve their hitting abilities. Designated hitters can focus more on their opposing pitcher and pick up on their style and trends. J.D. Martinez has had an excellent career as a designated hitter thus far. He has played 776 games as a designated hitter and has 534 RBIs (FanGraphs). Martinez pulls through for his team as a designated hitter, hitting with men on base to produce an effective offense.

Even though I put hard work into this experiment, the research still has many things that could be improved. Because FanGraphs only gave me data starting in 2002, I  was not able to  get a holistic view of the batting of pitchers. Pitchers could have had better batting statistics in the 1900s; however, this experiment  was unable to  tackle that. The COVID-19 pandemic in 2020 also screwed up the data since there were designated hitters in the National League.  However, it was a shorter season, and I believed the data would  not   have a  great  impact on  my findings.  Also, I could have used more advanced metrics like Weighted Runs Created Plus (wRC+) and Statcast data to provide more evidence.  Lastly, designated hitters  do not usually  bat in the number nine slot since they are  usually  better hitters than pitchers.  Therefore, they could bat in the cleanup spot where more men could be on base. Designated hitters could have more opportunities to receive an RBI, whereas pitchers who bat last would  most  likely bat with no one on base.

There are many more factors that I can consider in further investigation. Pinch hitters used to have  a large  impact in National League games as a team would strategize when it was best to input a pinch-hitter. Managers also had to research what batter would be best to put in those situations. If I had researched pinch hitters more, I would have compared them to designated hitters and see which position would provide a more effective offense. Stolen bases are another  large  part of the offense, and it would be amazing to see if there was a  large  difference in the number of bases a pitcher stole versus a designated hitter. Shohei Ohtani, a pitcher and designated hitter, is a  huge  outlier in my research.  In the 2023 season, he had a .304 average,  which was  tenth in the MLB, and 44 home runs,  which was  4th in the MLB.  Not only is he excellent at hitting, but he is also  an amazing  pitcher, as he has a career ERA of 3.01. These stats led him to another MVP season (“Shohei Ohtani Career Stats - MLB”). With the success of Ohtani, there may be more pitchers that can hit and hit for power as Ohtani inspires a new generation.  The designated hitter might go away once more if this  does happen, and a whole  new set of data might appear.

In conclusion, National League designated hitters produce more RBIs per game and strike out less often than National League pitchers.  However,  I cannot conclusively say that  National League designated hitters provide a better offense than National League pitchers.  Every batter needs to be productive at bat.  Designated hitters  seem to be  doing better than pitchers as they strike out less often and hit when men are in scoring  position .   As mentioned  previously , Ortiz has had a  very  successful career as a designated hitter.  However, Shohei Ohtani has proven that pitching and hitting well are not incompatible. Despite Ohtani, designated hitters are outperforming pitchers in the National League in specific areas when it comes to batting.  Therefore,  I believe  the National League made the correct decision in adopting the Designated Hitter rule, and I expect to keep seeing National League designated hitters for years to come.

American League is founded , 30 November 2021, https://www.history.com/this-day-in-history/american-league-founding-mlb. Accessed 22 December 2023.

Fangraphs , 24 June 2019, https://www.fangraphs.com/leaders/major-league?stats=bat&lg=nl&lg=al&qual=y&ind=0&pos=dh&type=c%2C4%2C6%2C-1%2C7%2C11%2C21&month=15&startdate=&enddate=&season1=1871&season=2023&team=0&v_cr=202301. Accessed 22 December 2023.

“All time home run leaders at each position.” MLB.com , 11 September 2022, https://www.mlb.com/news/all-time-home-run-leaders-at-each-position. Accessed 20 December 2023.

“Designated Hitter Rule | Glossary.” MLB.com , https://www.mlb.com/glossary/rules/designated-hitter-rule. Accessed 22 December 2023.

Kurkjian, Tim. “A little less 'magic': What we'll lose forever as pitchers stop hitting.” ESPN , 1 April 2022, https://www.espn.com/mlb/story/_/id/33621715/a-little-less-magic-mlb-lose-forever-pitchers-stop-hitting. Accessed 20 December 2023.

“Shohei Ohtani Career Stats - MLB.” ESPN , https://www.espn.com/mlb/player/stats/_/id/39832/category/pitching. Accessed 22 December 2023.

“What are the Differences Between the American and National Leagues?” Sports & Nobbies , 9 November 2023, https://www.sportsnhobbies.org/what-are-the-differences-between-the-american-and-national-leagues.htm. Accessed 22 December 2023.