“
different from 3.5. However, it is different from larger values, such as 4.0 (t = 2.89, df = 9, p = .019). Another example of this is provided in the Box 12.2. Finally, note that the one-sample t-test is identical to the paired-samples t-test for testing whether the mean D = 0. Indeed, the one-sample t-test for D = 0 produces the same results (t = 2.43, df = 9, p = .038). In Greater Depth … Box 12.2 Use of the T-Test in Performance Management: An Example Performance benchmarking is an increasingly popular tool in performance management. Public and nonprofit officials compare the performance of their agencies with performance benchmarks and draw lessons from the comparison. Let us say that a city government requires its fire and medical response unit to maintain an average response time of 360 seconds (6 minutes) to emergency requests. The city manager has suspected that the growth in population and demands for the services have slowed down the responses recently. He draws a sample of 10 response times in the most recent month: 230, 450, 378, 430, 270, 470, 390, 300, 470, and 530 seconds, for a sample mean of 392 seconds. He performs a one-sample t-test to compare the mean of this sample with the performance benchmark of 360 seconds. The null hypothesis of this test is that the sample mean is equal to 360 seconds, and the alternate hypothesis is that they are different. The result (t = 1.030, df = 9, p = .330) shows a failure to reject the null hypothesis at the 5 percent level, which means that we don’t have sufficient evidence to say that the average response time is different from the benchmark 360 seconds. We cannot say that current performance of 392 seconds is significantly different from the 360-second benchmark. Perhaps more data (samples) are needed to reach such a conclusion, or perhaps too much variability exists for such a conclusion to be reached. NONPARAMETRIC ALTERNATIVES TO T-TESTS The tests described in the preceding sections have nonparametric alternatives. The chief advantage of these tests is that they do not require continuous variables to be normally distributed. The chief disadvantage is that they are less likely to reject the null hypothesis. A further, minor disadvantage is that these tests do not provide descriptive information about variable means; separate analysis is required for that. Nonparametric alternatives to the independent-samples test are the Mann-Whitney and Wilcoxon tests. The Mann-Whitney and Wilcoxon tests are equivalent and are thus discussed jointly. Both are simplifications of the more general Kruskal-Wallis’ H test, discussed in Chapter 11.19 The Mann-Whitney and Wilcoxon tests assign ranks to the testing variable in the exact manner shown in Table 12.4. The sum of the ranks of each group is computed, shown in the table. Then a test is performed to determine the statistical significance of the difference between the sums, 22.5 and 32.5. Although the Mann-Whitney U and Wilcoxon W test statistics are calculated differently, they both have the same level of statistical significance: p = .295. Technically, this is not a test of different means but of different distributions; the lack of significance implies that groups 1 and 2 can be regarded as coming from the same population.20 Table 12.4 Rankings of
”
”
Evan M. Berman (Essential Statistics for Public Managers and Policy Analysts)