Problem set #9 Fall 2023

BIOL 4500 Name_________________________ Problem Set # 9 The city manager of a large city was negotiating with the three unions that represented the police, fire, and building inspectors over the salaries for these three groups of employees. The three unions claimed that the starting salaries were substantially different between the three groups, whereas in most cities there was not a significant difference in starting salaries between the three groups. To obtain information on starting salaries across the nation, the city manager decided to randomly select one city in each of eight geographical regions. The starting yearly salaries (in thousands of dollars) were obtained for each of the three groups in each of the eight regions. Using the data below, answer the following questions. Region 1 2 3 4 5 6 7 8 Police 32.3 33.2 30.8 30.5 30.1 30.2 28.4 27.9 Fire 31.9 32.8 31.6 31.2 30.8 30.6 28.7 27.5 Inspectors 27.9 27.8 26.5 26.8 26.4 26.8 25.3 25.9 (a) What are your null and alternative hypotheses? (2 pts.) Null: There was no difference in salaries for the 3 groups Alternative: There is a difference in the 3 groups per region. (b) Calculate means, standard deviation, and a 95% C.I. for each treatment. (2 pts.) Descriptive Statistics N Minimum Maximum Mean Std. Deviation Variance Police 8 27.90 33.20 30.4250 1.77261 3.142 Fire 8 27.50 32.80 30.6375 1.73776 3.020 Inspectors 8 25.30 27.90 26.6750 .87790 .771 Valid N (listwise) 8 One-Sample Test Test Value = 0 t df Significance Mean Difference 95% Confidence Interval o the Difference One-Sided p Two-Sided p Lower Upper Police 48.547 7 <.001 <.001 30.42500 28.9431 31.906 Fire 49.866 7 <.001 <.001 30.63750 29.1847 32.090

Inspector s 85.941 7 <.001 <.001 26.67500 25.9411 27.408 (c) Are these data are normally distributed? What evidence do you have of this? (2 pts.) Tests of Normality Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. Police .177 8 .200 * .954 8 .748 Fire .241 8 .189 .923 8 .453 Inspectors .193 8 .200 * .948 8 .688 *. This is a lower bound of the true significance. a. Lilliefors Significance Correction After testing for normality we can see the data are normal. All above .05 (d) Do these data meet the homogeneity of variance assumption for ANOVA? How do you know? What are your options if the variances are significantly different among groups? (2 pts.) Tests of Homogeneity of Variances Levene Statistic df1 df2 Sig. VAR00006 Based on Mean 1.153 2 21 .335 Based on Median .983 2 21 .391 Based on Median and with adjusted df .983 2 16.723 .395 Based on trimmed mean 1.110 2 21 .348 ANOVA Effect Sizes a Point Estimate 95% Confidence Interval Lower Upper VAR00006 Eta-squared .621 .280 .746 Epsilon-squared .585 .212 .722 Omega-squared Fixed- effect .574 .205 .713

There is differences as we can see in the sig. column, it tells us that the salaries have a difference, Meaning we can reject the null hypothesis. Multiple Comparisons Dependent Variable: VAR00006 (I) VAR00005 (J) VAR00005 Mean Difference (I- J) Std. Error Sig. 95% Confidence Interva Lower Bound Upper Bound Tukey HSD 1.00 2.00 -.2125 .76008 .958 -2.1283 1.703 3.00 3.7500 * .76008 <.001 1.8342 5.665 2.00 1.00 .2125 .76008 .958 -1.7033 2.128 3.00 3.9625 * .76008 <.001 2.0467 5.878 3.00 1.00 -3.7500 * .76008 <.001 -5.6658 -1.834 2.00 -3.9625 * .76008 <.001 -5.8783 -2.046 Bonferron i 1.00 2.00 -.2125 .76008 1.000 -2.1897 1.764 3.00 3.7500 * .76008 <.001 1.7728 5.727 2.00 1.00 .2125 .76008 1.000 -1.7647 2.189 3.00 3.9625 * .76008 <.001 1.9853 5.939 3.00 1.00 -3.7500 * .76008 <.001 -5.7272 -1.772 2.00 -3.9625 * .76008 <.001 -5.9397 -1.985 Based on observed means. The error term is Mean Square(Error) = 2.311. *. The mean difference is significant at the .05 level. (h) What are the effect size and statistical power for you test? (3 pts.) Power Analysis Table Power b Test Assumptions N c Std. Dev. Effect Size d Sig. Overall Test a .995 24 1.7 1.311 .05 a. Test the null hypothesis that population mean is the same for all groups. b. Based on noncentral F-distribution. c. Total sample size across groups. d. Effect size measured by the root-mean-square standardized effect.