91Ó°ÊÓ

Given the following ANOVA output, answer the questions that follow: $$ \begin{aligned} &\text { Analysis of Variance for Response }\\\ &\begin{array}{lrrrrr} \text { Source } & \text { df } & \text { SS } & \text { MS } & F & P \\ \text { Block } & 6 & 1712.37 & 285.39 & 134.20 & 0.000 \\ \text { Treatment } & 3 & 2.27 & 0.76 & 0.36 & 0.786 \\ \text { Error } & 18 & 38.28 & 2.13 & & \\ \text { Total } & 27 & 1752.91 & & & \end{array} \end{aligned} $$ (a) The researcher wants to test \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}\) against \(H_{1}:\) at least one of the means is different. Based on the ANOVA table, what should the researcher conclude? (b) What is the mean square due to error? (c) Explain why it is not necessary to use Tukey's test on these data.

The variability among the sample means is called _____ sample variability, and the variability of each sample is the _____ sample variability.

The following data are taken from four different populations that are known to be normally distributed, with equal population variances based on independent simple random samples. $$ \begin{array}{cccc} \text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } & \text { Sample 4 } \\ \hline 110 & 138 & 98 & 130 \\ \hline 85 & 140 & 100 & 116 \\ \hline 83 & 130 & 94 & 157 \\ \hline 95 & 115 & 110 & 137 \\ \hline 103 & 101 & 104 & 144 \\ \hline 105 & 130 & 118 & 124 \\ \hline 107 & 123 & 102 & 139 \\ \hline \end{array} $$ (a) Test the hypothesis that each sample comes from a population with the same mean at the \(\alpha=0.05\) level of significance. That is, test \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}\). (b) If you rejected the null hypothesis in part (a), use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05\). (c) Draw boxplots of each set of sample data to support your results from parts (a) and (b).

Determine the F-test statistic based on the given summary statistics. $$ \begin{array}{cccc} \text { Population } & \text { Sample Size } & \text { Sample Mean } & \text { Sample Variance } \\ \hline 1 & 10 & 40 & 48 \\ \hline 2 & 10 & 42 & 31 \\ \hline 3 & 10 & 44 & 25 \end{array} $$

Given the following ANOVA output, answer the questions that follow. \(\begin{array}{lrrrrr}\text { Source } & \text { df } & \text { SS } & \text { MS } & F & P \\ \text { Factor A } & 1 & 531.2 & 531.2 & 11.73 & 0.003 \\\ \text { Factor B } & 2 & 3018.0 & 1509.0 & 33.33 & 0.000 \\ \text { Interaction } & 2 & 16.3 & 8.2 & 0.18 & 0.836 \\ \text { Error } & 18 & 814.9 & 45.3 & & \end{array}\) (a) Is there evidence of an interaction effect? Why or why not? (b) Based on the \(P\) -value, is there evidence of a difference in the means from factor A? Based on the \(P\) -value, is there evidence of a difference in the means from factor \(\mathrm{B} ?\) (c) What is the mean square error?

The comparisonwise error rate, denoted \(\alpha_{c}\), is the probability of making a Type I error when comparing two means. It is related to the familywise error rate, \(\alpha\), through the formula \(1-\alpha=\left(1-\alpha_{c}\right)^{k},\) where \(k\) is the number of means being compared. (a) If the familywise error rate is \(\alpha=0.05\) and \(k=3\) means are being compared, what is the comparisonwise error rate? (b) If the familywise error rate is \(\alpha=0.05\) and \(k=5\) means are being compared, what is the comparisonwise error rate? (c) Based on the results of parts (a) and (b), what happens to the comparisonwise error rate as the number of means compared increases?