Estimation and Hypothesis Testing: Two Populations

Transcription

1 1 Excel Manual Estimation and Hypothesis Testing: Two Populations Chapter 10 Chapter 10 takes the last two chapters a little further. This time each problem contains two samples, so the statistics and the calculations involved are a little more complicated. Consequently, the spreadsheets to accomplish these tasks will be a bit more complex. This chapter looks at the calculation of various confidence intervals for the difference of two means and two proportions as well hypothesis testing on both. It also covers the idea of a matched pair design, where two samples are taken in such a way that each item in Sample 1 may be paired with an item in Sample 2 to derive the difference between the two. Inference About the Difference Between Two Population Means for Independent Samples: σ1 and σ2 Known Large Sample Confidence Interval This sheet is very similar to the one we created for the confidence interval in Chapter 8. Here, however, we need to enter two arguments for each statistic, and these statistics need to be combined to create an overall mean and standard deviation. These differences have been incorporated in the following spreadsheet. This sheet will help find the confidence interval desired. Note the combined standard error calculation in cell B9. Since the calculation of the standard error is not straightforward, the CONFIDENCE() function cannot be used.

2 2 Excel Manual As seen in Chapters 8 and 9, if you would like to use lists of data instead of the statistics, you may change the inputs (the white cells) to the appropriate formula to include the columns of data. To make the spreadsheet look a little neater, the cells that cover two columns were formatted using the Merge & Center feature in the Alignment group of the Home tab. For example, select cells B8:B9 and choose Merge & Center. The following spreadsheet shows the results of a confidence interval for the difference in average annual premiums for employer-sponsored health insurance for family coverage in 2011 and 2010, from Example 10-3 on page 465 in your text. The 250 employees sampled in the year 2011 had a and ; the 200 employees sampled in the year 2010 had a and. Thus, the 97% confidence interval for the true mean difference between the average annual premiums for employer-sponsored health insurance for family coverage in 2011 and 2010 is ($877.40, $ ). The values given on page 466 in the text are slightly different due to rounding. Hypothesis Testing on the Difference of Two Means Similar to the confidence interval calculations in the previous section, performing a hypothesis test requires combining the standard deviation to calculate the standard error and find the difference of the two sample means. Otherwise, the spreadsheet is very similar to the one created in Chapter 9. Again, if lists of data are used, replace the inputs with the appropriate function on the columns containing the data. The null hypothesis for a difference of two means is often 0 in other words, the assumption is that the two means are the same. You can test for the difference being something other than 0 by entering the hypothesized difference in the area provided in the spreadsheet in cell B7.

3 3 Excel Manual The following hypothesis test uses the same data as the above confidence interval, this time following Example 10-4 on page 467. The hypothesis being tested is that the credit card debt is different at the.01 significance level for the two age groups. The p-value for the two-tailed test is essentially 0. This is much less than the significance level of.01 indicated in the example, which means we reject the null hypothesis. There is a difference in the average annual premium for employer-sponsored health insurance for family coverage for 2011 and 2010.

4 4 Excel Manual Inference About the Difference Between Two Population Means for Independent Samples: σ1 and σ2 Unknown So far, the spreadsheets have been very similar. The next spreadsheet shows how the concepts may be combined in the same spreadsheet, since the inputs for both confidence intervals and hypothesis tests are the same and the calculations are similar. Only the output is different. The following example uses a pooled standard deviation, the calculation of which is in cell B9. The standard error is then calculated from this pooled standard deviation; it appears in cell B10. Example 10-5 on page 472 looks at the difference in the mean amounts of caffeine in two brands of coffee. A sample of 15 jars of Brand 1 had a mean of 80mg and a standard deviation of 5mg. A sample of 12 jars of Brand 2 had a mean of 77mg and a standard deviation of 6mg. A hypothesis test and confidence interval for the small sample difference in means test, assuming that the variance is equal (pooled) for both populations, are illustrated below:

5 5 Excel Manual The 95% confidence interval for the difference in mean caffeine amounts for the two brands of coffee is (-1.357, 7.357). Using the same data for a hypothesis test to test whether Brand 2 has more caffeine than Brand 1 is a one-tailed test using a less than alternative hypothesis since Brand 1 was entered on the left and Brand 2 on the right. The p-value is If α =.1, then the null hypothesis would be rejected. Any α =.05 or lower would fail to reject the null hypothesis. For this α, Brand 2 s caffeine level is not significantly higher than Brand 1. Inferences About the Difference Between Two Population Means for Independent Samples: σ1 and σ2 Unknown and Unequal This concept combines the previous two sheets. The calculation of the standard error comes from the large sample unequal variance sheet and the test-statistic use of the T-distribution from the small sample problem. Another difference is that the Degrees of Freedom will be the smaller sample size minus 1. The following example uses the same data as the previous examples, but this time it incorporates the small sample unequal variances formulas. The formula in B9 uses the nonpooled formula for the standard deviation of sample means. The calculation of the T α/2 and the p-value both use the different degrees of freedom calculation, utilizing the Satterthwaite correction to calculate the degrees of freedom since the sample size is small. This value is rounded down to get the degrees of freedom used in the test statistic and the critical test statistic for the confidence interval.

6 6 Excel Manual The results using the previous example s data (from Example 10-8 on page 482) with this correction for the degrees of freedom is shown below:

7 7 Excel Manual As you can see, the p-value changed slightly, from.085 to.089, making it slightly less significant. In addition, the confidence interval widened slightly to (-1.49, 7.49) from (-1.35, 7.35). Inference About the Difference Between Two Population Means for Paired Samples Matched pair inferences are calculated very similarly to the single sample methods studied earlier. The only difference is that the inferences are performed on the list of differences as opposed to the original data. To do these calculations, pick the appropriate single sample test that corresponds to the matched pair test and enter the statistics calculated on the matched pair differences. If the original data is used, a column may be set up for each measurement and a third column may be set up to hold the difference between the two. The statistics can then be calculated in this third column. Data Analysis ToolPak for Performing the Hypothesis Tests on Two Samples The following illustration shows the Data Analysis ToolPak s list of available options. Notice that the matched pair is available as well as two-sample with equal or unequal variances and the two sample test using the Z-test. Be aware that the options are limited compared to those illustrated in the text. For example, there is only a t-test available for a Matched Pair design. In addition, there is no option for a confidence interval for any of the two sample options. These can only be used with existing data. In other words, there is no way to enter derived statistics to obtain a result. The following material includes four sets of illustrations detailing the process involved in using the Data Analysis ToolPak. All use the same data from Exercise on page 508, which counts mosquito bites found after using two brands of mosquito repellents on eight guests at a fishing camp. Match Pair Two Sample t-test on Means The setup is shown on the left; results are illustrated on the right:

8 8 Excel Manual Two Sample t-test Mean with Equal Variances The setup is shown on the left; results are illustrated on the right:

9 9 Excel Manual Two Sample t-test Unequal Variances The setup is shown on the left; results are illustrated on the right: Two Sample Z-test for Means The setup is shown on the left; results are illustrated on the right: Inferences About the Difference Between Two Population Proportions for Large and Independent Samples The following example illustrates how to set up a spreadsheet to calculate the confidence interval for the difference between two population proportions. B7 contains the difference in the sample proportions; the calculation in cell B11 uses a non-pooled approach to calculating the standard error.

10 10 Excel Manual Example on page 498 in your text references a study that compares users of two different brands of toothpaste specifically, the proportion of each group who say they will never switch to another brand. We will create a 97% confidence interval for the difference in proportion of toothpaste Brand A users and toothpaste Brand B users who say they will never switch. The 97% confidence interval for the difference in proportion of toothpaste A users and toothpaste B users is ( , ).

11 11 Excel Manual Hypothesis Testing on Two Proportions For hypothesis testing on two proportions, the assumption in the null hypothesis is that the proportions are the same. Therefore, we assume the variances are the same for each sample. As a result, a pooled formula is used this time. The difference in this spreadsheet is that the combined proportion is calculated in B11 and the standard error for a pooled sample proportion is calculated in cell B12. Otherwise, this spreadsheet is very similar to the one built to find the confidence interval for two proportions. You may want to combine the calculation portion of this sheet and the sheet for the confidence interval into one, thus making a single sheet for calculating many statistics of a two proportions situation. In B11, enter the formula =(B5+C5)/(B4+C4) to calculate the combined proportion. In B12, enter the formula =SQRT((B11*(1-B11))*(1/B4+1/C4)) to calculate the pooled standard error for the two proportions. The rest of the spreadsheet is similar to previous spreadsheets. Example on page 499 in your text tests the hypothesis that the proportion of people who say they will never switch from toothpaste A is greater than the proportion who say they won t switch from toothpaste B, with a.01 significance level.

12 12 Excel Manual The results are shown below: The p-value for this test is This is greater than the significance level of.01 indicated for the test. Therefore, we fail to reject the null hypothesis. There is not enough evidence to say that people who use toothpaste A are more apt to not switch to another brand.