Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
STAT 500 Solutions Practice Problems: ANOVA 1. To compare the heights of Soprano, Alto, Tenor and Bass singers, data have been collected from the NY Choral Society in 1979 (singers.txt). Determine whether these four groups have the same mean height. Use α=0.05. (Source: DASL) We use ANOVA to answer this question as we have a response variable (heights) measured across a categorical variable (Singers) of more than two levels (Soprano, Alto, Tenor, and Bass). Ho: is that the mean heights of the four types of singers are equal. In statistical notation this is: µS = µA = µT = µB Ha: is that at least one of these means differ, or can be written as not all the means are equal. With a large sample size we are not overly concerned with normality, but since the sample sizes differ across the groups (36, 35, 20, and 39, respectively) we would be concerned if the variances were not equal. From Stat > ANOVA > Test for equal variances using “Response data are in separate column for each factor level” we get the following. This indicates a violation of the equal variance assumption and could allow one to question the reliability of our results. Test for Equal Variances: Soprano, Alto, Tenor, Bass Multiple comparison intervals for the standard deviation, α = 0.05 Multiple Comparisons P-Value Soprano 0.031 Levene’s Test P-Value 0.020 Alto Tenor Bass 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 If intervals do not overlap, the corresponding stdevs are significantly different. 1 To continue with the ANOVA analysis in Minitab selecting Stat > ANOVA > One-Way use choose that the “Response data are in separate column for each factor level.” We also check the Comparisons tab and check the box for Tukey. This results in the following output and multiple comparison graph. One-way ANOVA: Soprano, Alto, Tenor, Bass Method Null hypothesis Alternative hypothesis Significance level Rows unused All means are equal At least one mean is different α = 0.05 26 Equal variances were assumed for the analysis. Factor Information Factor Factor Levels 4 Values Soprano, Alto, Tenor, Bass Analysis of Variance Source Factor Error Total DF 3 126 129 Adj SS 1058.5 796.7 1855.3 Adj MS 352.843 6.323 F-Value 55.80 P-Value 0.000 Model Summary S 2.51462 R-sq 57.06% R-sq(adj) 56.03% R-sq(pred) 54.06% Means Factor Soprano Alto Tenor Bass N 36 35 20 39 Mean 64.250 64.886 69.150 70.718 StDev 1.873 2.795 3.216 2.361 95% (63.421, (64.045, (68.037, (69.921, CI 65.079) 65.727) 70.263) 71.515) Pooled StDev = 2.51462 Tukey Pairwise Comparisons Grouping Information Using the Tukey Method and 95% Confidence Factor Bass Tenor Alto Soprano N 39 20 35 36 Mean 70.718 69.150 64.886 64.250 Grouping A A B B Means that do not share a letter are significantly different. 2 Tukey Simultaneous 95% CIs Difference of Means for Soprano, Alto, ... Alto - Soprano Tenor - Soprano Bass - Soprano Tenor - Alto Bass - Alto Bass - Tenor 0 2 4 6 8 If an interval does not contain zero, the corresponding means are significantly different. From the ANOVA output F test statistic (55.8) and corresponding p-value (less than 0.001), we would reject the null hypothesis at a 0.05 level of significance. We would conclude that the sample data provides statistical evidence that there is a difference in the mean heights across the four types of singers. Since we have a significant result, we can review the multiple comparison results to determine where differences occur. From the letter grouping information, we see that Bass and Tenor are not different (both letter A) and Alto and Soprano are not different (both letter B). However, Bass and Tenor are different from Alto and Soprano since they do not have a common letter. From the multiple comparison graph of confidence intervals, we confirm the similarity between Alto and Soprano (interval includes zero) and Bass and Tenor (interval includes zero). Furthermore, we can reach the following conclusions in mean heights: Tenor exceeds Soprano; Bass exceeds Soprano; Tenor exceeds Alto; and Bass exceeds Alto. All of which reflects the results of the letter grouping. 3