Applying the Central Limit Theorem

Like any other set of numbers, this sampling distribution has a mean and a standard deviation. As is the case with the sampling distribution of the mean (see Chapters 9 and 10), the Central Limit Theorem applies here.

According to the Central Limit Theorem, if the samples are large, the sampling distribution of the difference between means is approximately a normal distribution. If the populations are normally distributed, the sampling distribution is a normal distribution even if the samples are small.

The Central Limit Theorem also has something to say about the mean and standard deviation of this sampling distribution. Suppose the parameters for the first population are μ₁ and σ₁, and the parameters for the second population are μ₂ and σ₂. The mean of the sampling distribution is

The standard deviation of the sampling distribution is

N₁ is the number of individuals in the sample from the first population, and N₂ is the number of individuals in the sample from the second.

This standard deviation is called the standard error of the difference between means.

Figure 11-2 shows the sampling distribution along with its parameters, as specified by the Central Limit Theorem.

Z’s once more

Because the Central Limit Theorem says that the sampling distribution is approximately normal for large samples (or for small samples from normally distributed populations), you use the z-score as your test statistic. Another way to say “Use the z-score as your test statistic” is “Perform a z-test.” Here’s the formula:

The term (μ₁-μ₂) represents the difference between the means in H₀.

This formula converts the difference between sample means into a standard score. Compare the standard score against a standard normal distribution — a normal distribution with μ = 0 and σ = 1. If the score is in the rejection region defined by α, reject H₀. If it’s not, don’t reject H₀.

You use this formula when you know the value of σ₁² and σ₂².

Here’s an example. Imagine a new training technique designed to increase IQ. Take a sample of 25 people and train them under the new technique. Take another sample of 25 people and give them no special training. Suppose that the sample mean for the new technique sample is 107, and for the no-training sample it’s 101.2. The hypothesis test is

H₀: μ₁ - μ₂ = 0

H₁: μ₁ - μ₂ > 0

I’ll set α at .05.

The IQ is known to have a standard deviation of 16, and I assume that standard deviation would be the same in the population of people trained on the new technique. Of course, that population doesn’t exist. The assumption is that if it did, it should have the same value for the standard deviation as the regular population of IQ scores. Does the mean of that (theoretical) population have the same value as the regular population? H₀ says it does. H₁ says it’s larger.

The test statistic is

With α = .05, the critical value of z — the value that cuts off the upper 5 percent of the area under the standard normal distribution — is 1.645. (You can use the worksheet function NORM.S.INV from Chapter 8 to verify this.) The calculated value of the test statistic is less than the critical value, so the decision is to not reject H₀. Figure 11-3 summarizes this.

Data analysis tool: z-Test: Two Sample for Means

Excel provides a data analysis tool that makes it easy to do tests like the one in the IQ example. It’s called z-Test: Two Sample for Means. Figure 11-4 shows the dialog box for this tool along with sample data that correspond to the IQ example.

To use this tool, follow these steps:

1. Type the data for each sample into a separate data array.

   For this example, the data in the New Technique sample are in column E, and the data for the No Training sample are in column G.

2. Select DATA | Data Analysis to open the Data Analysis dialog box.
3. In the Data Analysis dialog box, scroll down the Analysis Tools list and select z-Test: Two Sample for Means.
4. Click OK to open the z-Test: Two Sample for Means dialog box. (Refer to Figure 11-4.)

5. In the Variable 1 Range box, enter the cell range that holds the data for one of the samples.

   For the example, the New Technique data are in $E$2:$E$27. (Note the $ signs for absolute referencing.)

6. In the Variable 2 Range box, enter the cell range that holds the data for the other sample.

   The No Training data are in $G$2:$G$27.

7. In the Hypothesized Mean Difference box, type the difference between μ1 and μ2 that H₀ specifies.

   In this example, that difference is 0.

8. In the Variable 1 Variance (known) box, type the variance of the first sample.

   The standard deviation of the population of IQ scores is 16, so this variance is 16²= 256.

9. In the Variable 2 Variance (known) box, type the variance of the second sample.

   In this example, the variance is also 256.

10. If the cell ranges include column headings, select the Labels check box.

    I included the headings in the ranges, so I selected the box.

11. The Alpha box has 0.05 as a default.

    I used the default value, consistent with the value of α in this example.

12. In the Output Options, select a radio button to indicate where you want the results.

    I selected New Worksheet Ply to put the results on a new page in the worksheet.

13. Click OK.

    Because I selected New Worksheet Ply, a newly created page opens with the results.

Figure 11-5 shows the tool’s results, after I expanded the columns. Rows 4, 5, and 7 hold values you input into the dialog box. Row 6 counts the number of scores in each sample.

The value of the test statistic is in cell B8. The critical value for a one-tailed test is in B10, and the critical value for a two-tailed test is in B12.

Cell B9 displays the proportion of area that the test statistic cuts off in one tail of the standard normal distribution. Cell B11 doubles that value — it’s the proportion of area cut off by the positive value of the test statistic (in the tail on the right side of the distribution) plus the proportion cut off by the negative value of the test statistic (in the tail on the left side of the distribution).

Like peas in a pod: Equal variances

When you don’t know a population variance, you use the sample variance to estimate it. If you have two samples, you average (sort of) the two sample variances to arrive at the estimate.

Putting sample variances together to estimate a population variance is called pooling. With two sample variances, here’s how you do it:

In this formula, s_p² stands for the pooled estimate. Notice that the denominator of this estimate is (N₁-1) + (N₂-1). Is this the df? Absolutely!

The formula for calculating t is

On to an example. FarKlempt Robotics is trying to choose between two machines to produce a component for its new microrobot. Speed is of the essence, so the company has each machine produce ten copies of the component and they time each production run. The hypotheses are

H₀: μ₁-μ₂ = 0

H₁: μ₁-μ₂ ≠ 0

They set α at .05. This is a two-tailed test because they don’t know in advance which machine might be faster.

Table 11-1 presents the data for the production times in minutes.

The pooled estimate of σ² is

The estimate of σ is 2.75, the square root of 7.56.

The test statistic is

For this test statistic, df = 18, the denominator of the variance estimate. In a t-distribution with 18 df, the critical value is 2.10 for the right-side (upper) tail and –2.10 for the left-side (lower) tail. If you don’t believe me, apply T.INV.2T (see Chapter 10). The calculated value of the test statistic is greater than 2.10, so the decision is to reject H₀. The data provide evidence that Machine 2 is significantly faster than Machine 1. (You can use the word significant whenever you reject H₀.)

Like p’s and q’s: Unequal variances

The case of unequal variances presents a challenge. As it happens, when variances are not equal, the t-distribution with (N₁-1) + (N₂-1) degrees of freedom is not as close an approximation to the sampling distribution as statisticians would like.

Statisticians meet this challenge by reducing the degrees of freedom. To accomplish the reduction, they use a fairly involved formula that depends on the sample standard deviations and the sample sizes.

Because the variances aren’t equal, a pooled estimate is not appropriate. So you calculate the t-test in a different way:

You evaluate the test statistic against a member of the t-distribution family that has the reduced degrees of freedom.

T.TEST

The worksheet function T.TEST eliminates the muss, fuss, and bother of working through the formulas for the t-test.

Figure 11-6 shows the data for the FarKlempt machines example I show you earlier in the chapter. The figure also shows the Function Arguments dialog box for T.TEST.

Follow these steps:

1. Type the data for each sample into a separate data array and select a cell for the result.

   For this example, the data for the Machine 1 sample are in column B and the data for the Machine 2 sample are in column D.

2. From the Statistical Functions menu, select T.TEST to open the Function Arguments dialog box for T.TEST.

3. In the Function Arguments dialog box, enter the appropriate values for the arguments.

   In the Array1 box, enter the sequence of cells that holds the data for one of the samples.

   In this example, the Machine 1 data are in B3:B12.

   In the Array2 box, enter the sequence of cells that holds the data for the other sample.

   The Machine 2 data are in D3:D12.

   The Tails box indicates whether this is a one-tailed test or a two-tailed test. In this example, it’s a two-tailed test, so I typed 2 in this box.

   The Type box holds a number that indicates the type of t-test. The choices are 1 for a paired test (which you find out about in an upcoming section), 2 for two samples assuming equal variances, and 3 for two samples assuming unequal variances. I typed 2.

   With values supplied for all the arguments, the dialog box shows the probability associated with the t value for the data. It does not show the value of t.

4. Click OK to put the answer in the selected cell.

The value in the dialog box in Figure 11-6 is less than .05, so the decision is to reject H₀.

By the way, for this example, typing 3 into the Type box (indicating unequal variances) results in a very slight adjustment in the probability from the equal variance test. The adjustment is small because the sample variances are almost equal and the sample sizes are the same.

Data analysis tool: t-Test: Two Sample

Excel provides data analysis tools that carry out t-tests. One tool works for the equal variance cases, and another for the unequal variances case. As you’ll see, when you use these tools, you end up with more information than T.TEST gives you.

Here’s an example that applies the equal variances t-test tool to the data from the FarKlempt machines example. Figure 11-7 shows the data along with the dialog box for t-Test: Two-Sample Assuming Equal Variances.

To use this tool, follow these steps:

1. Type the data for each sample into a separate data array.

   For this example, the data in the Machine 1 sample are in column B and the data for the Machine 2 sample are in column D.

2. Select DATA | Data Analysis to open the Data Analysis dialog box.
3. In the Data Analysis dialog box, scroll down the Analysis Tools list and select t-Test: Two Sample Assuming Equal Variances.

4. Click OK to open this tool’s dialog box.

   This is the dialog box in Figure 11-7.

5. In the Variable 1 Range box, enter the cell range that holds the data for one of the samples.

   For the example, the Machine 1 data are in $B$2:$B$12, including the column heading. (Note the $ signs for absolute referencing.)

6. In the Variable 2 Range box, enter the cell range that holds the data for the other sample.

   The Machine 2 data are in $D$2:$D$12, including the column heading.

7. In the Hypothesized Mean Difference box, type the difference between μ1 and μ2 that H₀ specifies.

   In this example, that difference is 0.

8. If the cell ranges include column headings, select the Labels check box.

   I included the headings in the ranges, so I selected the box.

9. The Alpha box has 0.05 as a default. Change that value if you’re so inclined.

10. In the Output Options, select a radio button to indicate where you want the results.

    I selected New Worksheet Ply to put the results on a new page in the worksheet.

11. Click OK.

    Because I selected New Worksheet Ply, a newly created page opens with the results.

Figure 11-8 shows the tool’s results, after I expanded the columns. Rows 4 through 7 hold sample statistics. Cell B8 shows the H₀-specified difference between the population means, and B9 shows the degrees of freedom.

The remaining rows provide t-related information. The calculated value of the test statistic is in B10. Cell B11 gives the proportion of area that the positive value of the test statistic cuts off in the upper tail of the t-distribution with the indicated df. Cell B12 gives the critical value for a one-tailed test: That’s the value that cuts off the proportion of the area in the upper tail equal to α.

Cell B13 doubles the proportion in B11. This cell holds the proportion of area from B11 added to the proportion of area that the negative value of the test statistic cuts off in the lower tail. Cell B14 shows the critical value for a two-tailed test: That’s the positive value that cuts off α/2 in the upper tail. The corresponding negative value (not shown) cuts off α/2 in the lower tail.

The samples in the example have the same number of scores and approximately equal variances, so applying the unequal variances version of the t-Test tool to that data set won’t show much of a difference from the equal variances case.

Instead I created another example, summarized in Table 11-2. The samples in this example have different sizes and widely differing variances.

To show you the difference between the equal variances tool and the unequal variances tool, I ran both on the data and put the results side by side. Figure 11-9 shows the results from both tools. To run the Unequal Variances tool, you complete the same steps as for the Equal Variances version, with one exception: In the Data Analysis Tools dialog box, you select t-Test: Two Sample Assuming Unequal Variances.

Figure 11-9 shows one obvious difference between the two tools: The Unequal Variances tool shows no pooled estimate of σ², because the t-test for that case doesn’t use one. Another difference is in the df. As I point out earlier, in the unequal variances case, you reduce the df based on the sample variances and the sample sizes. For the equal variances case, the df in this example is 12, and for the unequal variances case, it’s 10.

The effects of these differences show up in the remaining statistics. The t values, critical values, and probabilities are different.

T.TEST for matched samples

Earlier, I describe the worksheet function T.TEST and show you how to use it with independent samples. This time, I use it for the matched samples weight-loss example. Figure 11-10 shows the Function Arguments dialog box for T.TEST along with data from the weight-loss example.

Here are the steps to follow:

1. Enter the data for each sample into a separate data array and select a cell.

   For this example, the data for the Before sample are in column B and the data for the After sample are in column C.

2. From the Statistical Functions menu, select T.TEST to open the Function Arguments dialog box for T.TEST.

3. In the Function Arguments dialog box, enter the appropriate values for the arguments.

   In the Array1 box, type the sequence of cells that holds the data for one of the samples. In this example, the Before data are in B3:B12.

   In the Array2 box, type the sequence of cells that holds the data for the other sample.

   The After data are in C3:C12.

   The Tails box indicates whether this is a one-tailed test or a two-tailed test. In this example, it’s a one-tailed test, so I type 1 in the Tails box.

   The Type box holds a number that indicates the type of t-test to perform. The choices are 1 for a paired test, 2 for two samples assuming equal variances, and 3 for two samples assuming unequal variances. I typed 1.

   With values supplied for all the arguments, the dialog box shows the probability associated with the t value for the data. It does not show the value of t.

4. Click OK to put the answer in the selected cell.

The value in the dialog box in Figure 11-10 is less than .05, so the decision is to reject H₀.

If I assign the column headers in Figure 11-10 as names for the respective arrays, the formula in the Formula bar can be

=T.TEST(Before,After,1,1)

That format might be easier to explain if you had to show the worksheet to someone. (If you don’t remember how to define a name for a cell range, refer to Chapter 2.)

Data analysis tool: t-Test: Paired Two Sample for Means

Excel provides a data analysis tool that takes care of just about everything for matched samples. It’s called t-test: Paired Two Sample for Means. In this section, I use it on the weight-loss data.

Figure 11-11 shows the data along with the dialog box for t-Test: Paired Two Sample for Means.

Here are the steps to follow:

1. Enter the data for each sample into a separate data array.

   For this example, the data in the Before sample are in column B and the data for the After sample are in column C.

2. Select Data | Data Analysis to open the Data Analysis dialog box.
3. In the Data Analysis dialog box, scroll down the Analysis Tools list, and select t-Test: Paired Two Sample for Means.

4. Click OK to open this tool’s dialog box.

   This is the dialog box in Figure 11-11.

5. In the Variable 1 Range box, enter the cell range that holds the data for one of the samples.

   For the example, the Before data are in $B$2:$B$12, including the heading. (Note the $ signs for absolute referencing.)

6. In the Variable 2 Range box, enter the cell range that holds the data for the other sample.

   The After data are in $C$2:$C$12, including the heading.

7. In the Hypothesized Mean Difference box, type the difference between μ₁ and μ₂ that H₀ specifies.

   In this example, that difference is 0.

8. If the cell ranges include column headings, select the Labels check box.

   I included the headings in the ranges, so I selected the check box.

9. The Alpha box has 0.05 as a default. Change that value if you want to use a different α.

10. In the Output Options, select a radio button to indicate where you want the results.

    I selected New Worksheet Ply to put the results on a new page in the worksheet.

11. Click OK.

    Because I selected New Worksheet Ply, a newly created page opens with the results.

Figure 11-12 shows the tool’s results, after I expanded the columns. Rows 4 through 7 hold sample statistics. The only item that’s new is the number in cell B7, the Pearson Correlation Coefficient. This is a number between –1 and +1 that indicates the strength of the relationship between the data in the first sample and the data in the second.

If this number is close to 1 (as in the example), high scores in one sample are associated with high scores in the other, and low scores in one are associated with low scores in the other. If the number is close to –1, high scores in the first sample are associated with low scores in the second, and low scores in the first are associated with high scores in the second.

If the number is close to zero, scores in the first sample are unrelated to scores in the second. Because the two samples consist of scores on the same people, you expect a high value. (I describe this topic in much greater detail in Chapter 15.)

Cell B8 shows the H₀-specified difference between the population means, and B9 shows the degrees of freedom.

The remaining rows provide t-related information. The calculated value of the test statistic is in B10. Cell B11 gives the proportion of area the positive value of the test statistic cuts off in the upper tail of the t-distribution with the indicated df. Cell B12 gives the critical value for a one-tailed test: That’s the value that cuts off the proportion of the area in the upper tail equal to α.

Cell B13 doubles the proportion in B11. This cell holds the proportion of area from B11 added to the proportion of area that the negative value of the test statistic cuts off in the lower tail. Cell B13 shows the critical value for a two-tailed test: That’s the positive value that cuts off α/2 in the upper tail. The corresponding negative value (not shown) cuts off α/2 in the lower tail.

Using F in conjunction with t

One use of the F-distribution is in conjunction with the t-test for independent samples. Before you do the t-test, you use F to help decide whether to assume equal variances or unequal variances in the samples.

In the equal variances t-test example I show you earlier, the standard deviations are 2.71 and 2.79. The variances are 7.34 and 7.78. The F-ratio of these variances is

Each sample is based on ten observations, so df = 9 for each sample variance. An F-ratio of 1.06 cuts off the upper 47 percent of the F-distribution whose df are 9 and 9, so it’s safe to use the equal-variances version of the t-test for these data.

In the sidebar at the end of Chapter 10, I mention that on rare occasions a high α is a good thing. When H₀ is a desirable outcome and you’d rather not reject it, you stack the deck against rejecting by setting α at a high level so that small differences cause you to reject H₀.

This is one of those rare occasions. It’s more desirable to use the equal variances t-test, which typically provides more degrees of freedom than the unequal variances t-test. Setting a high value of α (.20 is a good one) for the F-test enables you to be confident when you assume equal variances.

F.TEST

The worksheet function F.TEST calculates an F-ratio on the data from two samples. It doesn’t return the F-ratio. Instead, it provides the two-tailed probability of the calculated F-ratio under H₀. This means that the answer is the proportion of area to the right of the F-ratio, and to the left of the reciprocal of the F-ratio (1 divided by the F-ratio).

Figure 11-14 presents the data for the FarKlempt machines example I just summarized for you. The figure also shows the Function Arguments dialog box for F.TEST.

Follow these steps:

1. Enter the data for each sample into a separate data array and select a cell for the answer.

   For this example, the data for the Machine 1 sample are in column B and the data for the Machine 2 sample are in column D.

2. From the Statistical Functions menu, select F.TEST to open the Function Arguments dialog box for F.TEST.

3. In the Function Arguments dialog box, enter the appropriate values for the arguments.

   In the Array1 box, enter the sequence of cells that holds the data for the sample with the larger variance. In this example, the Machine 1 data are in B3:B12.

   In the Array2 box, enter the sequence of cells that holds the data for the other sample. The Machine 2 data are in D3:D17.

   With values entered for all the arguments, the answer appears in the dialog box.

4. Click OK to put the answer in the selected cell.

The value in the dialog box in Figure 11-14 is greater than .05, so the decision is to not reject H₀. Figure 11-15 shows the area that the answer represents.

Had I assigned names to those two arrays, the formula in the Formula bar could have been

=F.TEST(Machine_1,Machine_2)

If you don’t know how to assign names to arrays, see Chapter 2. In that chapter, you also find out why I inserted an underscore into each name.

F.DIST and F.DIST.RT

You use the worksheet function F.DIST or the function F.DIST.RT to decide whether or not your calculated F-ratio is in the region of rejection. For F.DIST, you supply a value for F, a value for each df, and a value (TRUE or FALSE) for an argument called Cumulative. If the value for Cumulative is TRUE, F.DIST returns the probability of obtaining an F-ratio of at most as high as yours if H₀ is true. (Excel calls this the left-tail probability.) If that probability is greater than 1- α, you reject H₀. If the value for Cumulative is FALSE, F.DIST returns the height of the F-distribution at your value of F. I use this option later in this chapter to create a chart of the F-distribution.

F.DIST.RT returns the probability of obtaining an F-ratio at least as high as yours if H₀ is true. (Excel calls this the right-tail probability.) If that value is less than α, reject H₀. In practice, F.DIST.RT is more straightforward.

Here, I apply F.DIST.RT to the preceding example. The F-ratio is 1.36, with 9 and 14 df.

The steps are:

1. Select a cell for the answer.
2. From the Statistical Functions menu, select F.DIST.RT to open the Function Arguments dialog box for F.DIST.RT. (See Figure 11-16.)

3. In the Function Arguments dialog box, enter the appropriate values for the arguments.

   In the X box, type the calculated F. For this example, the calculated F is 1.36.

   In the Deg_freedom1 box, type the degrees of freedom for the variance estimate in the numerator of the F. The degrees of freedom for the numerator in this example is 9 (10 scores – 1).

   In the Deg_freedom2 box, I type the degrees of freedom for the variance estimate in the denominator of the F.

   The degrees of freedom for the denominator in this example is 14 (15 scores – 1).

   With values entered for all the arguments, the answer appears in the dialog box.

4. Click OK to close the dialog box and put the answer in the selected cell.

The value in the dialog box in Figure 11-16 is greater than .05, so the decision is to not reject H₀.

F.INV and F.INV.RT

The F.INV worksheet functions are the reverse of the F.DIST functions. F.INV finds the value in the F-distribution that cuts off a given proportion of the area in the lower (left) tail. F.INV.RT finds the value that cuts off a given proportion of the area in the upper (right) tail. You can use F.INV.RT to find the critical value of F.

Here, I use F.INV.RT to find the critical value for the two-tailed test in the FarKlempt machines example:

1. Select a cell for the answer.
2. From the Statistical Functions menu, select F.INV.RT to open the Function Arguments dialog box for FINV.RT.

3. In the Function Arguments dialog box, enter the appropriate values for the arguments.

   In the Probability box, enter the proportion of area in the upper tail. In this example, that’s .025 because it’s a two-tailed test with α = .05.

   In the Deg_freedom1 box, type the degrees of freedom for the numerator. For this example, df for the numerator = 9.

   In the Deg_freedom2 box, type the degrees of freedom for the denominator. For this example, df for the denominator = 14.

   With values entered for all the arguments, the answer appears in the dialog box. (See Figure 11-17.)

4. Click OK to put the answer into the selected cell.

Data analysis tool: F-test: Two Sample for Variances

Excel provides a data analysis tool for carrying out an F-test on two sample variances. I apply it here to the sample variances example I’ve been using. Figure 11-18 shows the data, along with the dialog box for F-Test: Two-Sample for Variances.

To use this tool, follow these steps:

1. Enter the data for each sample into a separate data array.

   For this example, the data in the Machine 1 sample are in column B and the data for the Machine 2 sample are in column D.

2. Select Data | Data Analysis to open the Data Analysis dialog box.
3. In the Data Analysis dialog box, scroll down the Analysis Tools list and select F-Test Two Sample For Variances.

4. Click OK to open this tool’s dialog box.

   This is the dialog box shown in Figure 11-18.

5. In the Variable 1 Range box, enter the cell range that holds the data for the first sample.

   For the example, the Machine 1 data are in $B$2:$B$12, including the heading. (Note the $ signs for absolute referencing.)

6. In the Variable 2 Range box, enter the cell range that holds the data for the second sample.

   The Machine 2 data are in $D$2:$D$17, including the heading.

7. If the cell ranges include column headings, select the Labels check box.

   I included the headings in the ranges, so I selected the box.

8. The Alpha box has 0.05 as a default. Change that value for a different α.

   The Alpha box provides a one-tailed alpha. I want a two-tailed test, so I changed this value to .025.

9. In the Output Options, select a radio button to indicate where you want the results.

   I selected New Worksheet Ply to put the results on a new page in the worksheet.

10. Click OK.

    Because I selected New Worksheet Ply, a newly created page opens with the results.

Figure 11-19 shows the tool’s results, after I expanded the columns. Rows 4 through 6 hold sample statistics. Cell B7 shows the degrees of freedom.

The remaining rows present F-related information. The calculated value of F is in B8. Cell B9 gives the proportion of area the calculated F cuts off in the upper tail of the F-distribution. This is the right-side area shown earlier in Figure 11-15. Cell B10 gives the critical value for a one-tailed test: That’s the value that cuts off the proportion of the area in the upper tail equal to the value in the Alpha box.