Statics

Week 2 – Lecture 3

Making judgements about differences between group statistics is one of the most powerful things that statistics can do for us. It is also one of the most counter-intuitive things that we need to master in the class.

Lecture 1 introduced the hypothesis testing procedure used in statistical testing. Lecture 2 examined how to set up, perform, and interpret the F test for variance equality. This lecture will focus on t-tests for testing mean equality. Again, these examples will use the compa-ratio variable, while the homework should use the Salary variable.

The T-Test

While we test for variance equality with an F test, we use the T-Test to test for mean equality testing. The t-test also uses the degree of freedom (df) value in providing us with our probability result; but again, Excel does the work for us.

There are three versions of the T-Test done for us by Excel. The first two are similar except one version is done if the variances are equal and the other if the variances are not equal. (Now we see the second reason for performing the F-test first.)

The third version of the T-test is for paired data, and is called T-test Paired Two Sample for Means. Paired data are two measures taken on the same subject. Examples include a math and English test score for students, preference for different drinks, and, in our data set the salary and midpoint values. Note that paired data must be measured in the same units, and be from the same subjects. Students in the past have incorrectly used the paired t-test on male and female salaries. These are not paired, as the measures are taken on different people and cannot be paired together for analysis.

In many ways, setting up Excel’s T-tests, and virtually all the functions we will study, follow the same steps as we just went through:

1. Set up the data into distinct groups. 2. Select the test function from either the Fx or Analysis list 3. Input the data ranges and output ranges into the appropriate entry boxes, checking

Labels if appropriate. 4. Clicking on OK to produce the output.

As with the F-test, the T-test has a couple of options depending upon what you want your output to look like. The Fx (or Formulas) option returns simply the p-value for the selected version of the test. The Data | Analysis selection provides descriptive statistics that are useful for additional analysis (some of which we will discuss later in the course).

The t-test requires that we select between three versions, one assuming equal variances between the populations, one assuming unequal variances in the populations, and one requiring paired data (two measures on each element in the sample, such as salary and midpoint for each person in our data set.) All have the same data set-up approach, so only one will be shown.

Question 2

The second question for this week asks about salary mean equality between males and females. The data and test set-up were discussed in Lecture two. Here is how we set up the steps for this question using compa-ratio as our example. (Note, the comments about each step made in the variance example above apply to this example as well, but they will not be repeated except for specific information related to the t-test outcome.) Specific differences from the variance example of question 1 will be highlighted with italics. Again, the results discussed with each step are shown in Lecture 2.

Step 1: Ho: Male compa-ratio mean = Female compa-ratio mean

Ha: Male compa-ratio mean =/= Female compa-ratio mean

Step 2: Alpha = 0.05

Step 3: t statistic and t-test, assuming equal variances. We use these as they are designed to test mean equality, and we are assuming equal variances.

Step 4: Reject the null hypothesis if the p-value is < alpha = 0.05.

Step 5: Perform the analysis. This step provides the cell location to place your results in, K35 for this question.

Setting up the data and test for question 2 about mean equality is similar to what one for the F- test question, and we can actually use the same data columns as we used in question 1 on variances. Again, after sorting the data into your comparison groups (with labels as we did for the F-test), select the appropriate test from either the Fx or Analysis list. A completed T-test Two-Sample Assuming Equal Variances input table is shown below.

The input box looks a lot like the one we saw for the F-test, and is completed in the same way. Enter the data ranges in the same order you have them listed in the hypothesis statements, check

the labels box if appropriate, and identify your output range top left cell (this is given in the homework problems for a consistent format for instructor grading). There is one input that differs and which we have not discussed, Hypothesized Mean Difference. For the most part, we do not use this. An example of when we might want to is when we have made a change and want to test its effectiveness. For example, we might have a pre- and post- test in a training course. In the original design, the average improvement might be 10 points on the post-test. If we change the design of the training, we would be interested not only in showing a significant change between the two tests but also a better change due to the revision. In this case, the first 10-point difference in the tests is a given, we want to know if the additional score change is significant. So, we enter 10 in the HMD box, and the analysis looks at only the mean difference larger than 10, the marginal improvement due to the design change. The input for the Fx T.Test contains 4 boxes, and produces the p-value in the cell the cursor is in. The first two boxes are the data range for each variable, and these should not have a label included. The third box asks whether you have a one or two tail test. The forth box asks

for the kind of test, paired, equal variance, or unequal variance.

Once we click OK for the T.test. we get a output, the p-value. When we click OK on the Analysi ToolPak function we get a more descriptive table; much like the differences with the two versions of the F. Here is a screen print for question 2 in this week’s homework. The question in the homework is about the equality of male and female salaries. For this example, we will look at the equality of male and female compa-ratios. Since we found (above) that the variances were not statistically different in the population, we will use the equal variance version of the 2- sample t-test.

Here is the screen shot for the result of the above data entry for a test about equal compa-ratio means for males and females.

As with the F-test output, the t-test starts with the test name, the group names, and some descriptive statistics. Line 4 start with a new result, the Pooled Variance; this is a weighted average of the sample variances since we are assuming that the related population variances are equal. The next line, hypothesized Mean Difference, shows up only if a value was entered in the data input box setting up the test.

Next comes our friend degrees of freedom, which equal the sum of both sample sizes minus 2 (or N1-1 + N2 -1). The calculated T value (similar to the calculated F value in the F test output) comes next. Note that since we have a negative t Stat, it falls in the left tail of the t distribution. This is important in one tail tests but not in two tail tests. Following the calculated T value come the one and two tail decision points. The one tail p-value is found in the P(T<==t) one-tail row followed by the T critical one-tail value. The two tail results follow.

Step 6: Conclusions and Interpretation. This step has several parts.

What is the p-value? 0.571 (our rounded compa-ratio example result).

(Since we again have a two-tail test, we use the P(T<=t) two-tail result.)

What is your decision: REJ or Not reject the null? NOT Rej (our compa-ratio result)

Why? The p-value (0.571) is > (greater than) 0.05. (The compa-ratio result)

What is your conclusion about the means in the population for the male and female salaries? We do not have enough evidence to say that the means differ in the population. The means are equal in the population. (Our compa-ratio result.)

Question 3

The third question for this week asks about salary differences based on educational level rather than gender. Here is how we set up the steps for this question using compa-ratio as our example. (Note, the comments about each step made in the variance example above apply to this example as well, but they will not be repeated except for specific information related to the t- test outcome.)

Note that this question has a directional focus (do employees with an advanced degree (degree code = 1) have higher average salaries?). This means we must develop a direction set of hypothesis statements. We will use the terms UnderG (for undergraduate degree code 0) and Grad (for graduate degree code 1) in these statements. Again, the results discussed with each step are shown in Lecture 2.

Step 1: Ho: UnderG mean compa-ratio => Grad mean compa-ratio

Ha: UnderG mean ratio < Grad mean compa-ratio

(Note the way the inequalities are set up; since the question is degree 1 salary means >, the question becomes the alternate hypothesis as it does not contain an = claim. These can be written in other ways, but must have the alternate relating to the question asked.)

Step 2: Alpha = 0.05

Step 3: t statistic and t-test assuming equal variances. We use these as they are designed to test mean equality. (The variance equality assumption is part of the question set-up.)

Step 4: Reject the null hypothesis if the p-value is < alpha = 0.05.

Step 5: Perform the analysis. This step provides the cell location to place your results in, K60 for this question (I60 for compa-ratio example). Except for using Degree as the sorting group variable, the data set-up for the third question is the same as done in question 2, so no new screen shot is needed.

Here is a screen print for a T-test on the question of whether the graduate and undergraduate degree compa-ratios means in the population are equal or not. We are assuming equal variances, we are using the T-Test Two-Sample Assuming Equal Variances form.

t-Test: Two-Sample Assuming Equal Variances

UnderG Grad Mean 1.05172 1.07324 Variance 0.00581 0.005999 Observations 25 25

Pooled Variance 0.005904

Hypothesized Mean

Difference 0 df 48 t Stat -0.99016 P(T<=t) one-tail 0.163529 t Critical one-tail 1.677224 P(T<=t) two-tail 0.327059 t Critical two-tail 2.010635

The table output is read exactly the same way as with the question 2 table with the exception that we are interested in the one-tail outcome, so we use the (highlighted) one-tail p-value row in our decision making.

Step 6: Conclusions and Interpretation. This step has several parts.

What is the p-value? 0.164(rounded) (our rounded compa-ratio example result).

(Since we again have a one-tail test, we use the P(T<=t) one-tail result.)

Is the t value in the t-distribution tail indicated by the arrow in the Ha claim? Yes. The t- value is negative, and the Ha arrow points to the left (or negative) tail of the t distribution.

(Since we only care about a difference in one direction, the result must be consistent with the desired direction. Only large negative values are of interest in this case, since our difference is calculated by (UnderG – Grad); large negative values show a larger Grad salary.)

What is your decision: REJ or Not reject the null? NOT Rej (our compa-ratio result)

Why? The p-value is > (greater than) 0.05. So, the sign does not matter in this case, but it is in the correct or negative tail. (The compa-ratio result)

What is your conclusion about the impact of education on average salaries? We do not have enough information to suggest that graduate degree holders have a higher average salary than undergraduate degree holders. (Our compa-ratio result.)

Special Case: The One-Sample T-test

Often, we may want to test the results of a sample against a standard; for example, is the weight of a production run of 8 ounces of canned pears actually equal to the standard of 8.02 oz.? (Note, most manufactures will put in slightly more than the label says to avoid being underweight which could result in a fine.)

Excel is not set up to perform this test, but we can “trick” it to do this for us. In the one- sample case, we need two pieces of information, the sample values and our comparison standard. Set these up as if they were any two-sample data sets, have our sample values (for example, 25 female compa-ratios in one column) and our comparison value in another. The comparison data column will only contain a single value equal to our comparison value. For example, we might want to test if the average female compa-ratio was greater than the compa-ratio midpoint of 1.00. The null would be H0: female compa-ratio mean <= 1.00 while the alternate would be Ha: Female compa-ratio mean > 1.00. The Compa-ratio data column would contain the Female compa-ratios and the other column (named for convenience as Ho Data) would contain only the value of 1.00, our standard value.

While we will leave the math for any interested student to perform, if we take the T-test unequal variance formulas for both the t-value and the df value and have a variance of 0 for one variable, both will reduce to the one-sample t-test formula and df value. Knowing this, we can use the unequal variance version of the t-test to perform what is essentially a one-sample test for us. Here is a screen shot of the data set-up and the t-test set-up window.

The output of this test will show a mean of 1.0 and a variance of 0 for the Ho Data (comparison) value, and the correct values for the Female compa-ratio variable, including the p- values.

Here is a video on setting up and using the t-test in Excel: https://screencast-o- matic.com/watch/cb6lYcImnn

Question 4

The forth question asks us to consider our findings. For the homework, you need to consider both the results found in the lectures as well as the results from your analysis of the salary data.

For our compa-ratio findings, it does not appear that the Equal Pay Act is being violated. From the sample of employees, it appears that males and females are being paid about the same relative to their grade midpoints across the grades. Last week showed about equal means and variances between the two groups, and this was confirmed with the

Please ask your instructor if you have any questions about this material.

When you have finished with this lecture, please respond to Discussion thread 3 for this week with your initial response and responses to others over a couple of days before reading the third lecture for the week.