91Ó°ÊÓ

Suppose you wish to compare the means of four populations based on independent random samples, each of which contains six observations. Insert, in an ANOVA table, the sources of variation and their respective degrees of freedom.

These data are observations collected using a completely randomized design: $$ \begin{array}{lll} \text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } \\ \hline 3 & 4 & 2 \\ 2 & 3 & 0 \\ 4 & 5 & 2 \\ 3 & 2 & 1 \\ 2 & 5 & \end{array} $$ a. Calculate CM and Total SS. b. Calculate SST and MST. c. Calculate SSE and MSE d. Construct an ANOVA table for the data. e. State the null and alternative hypotheses for an analysis of variance \(F\) -test. f. Use the \(p\) -value approach to determine whether there is a difference in the three population means.

The calcium content of a powdered mineral substance was analyzed five times by each of three methods, with similar standard deviations: $$ \begin{array}{llllll} \text { Method } & {\text { Percent Calcium }} \\ \hline 1 && .0279 & .0276 & .0270 & .0275 & .0281 \\ 2 && .0268 & .0274 & .0267 & .0263 & .0267 \\ 3 && .0280 & .0279 & .0282 & .0278 & .0283 \end{array} $$ Use an appropriate test to compare the three methods of measurement. Comment on the validity of any assumptions you need to make.

The data that follow are observations collected from an experiment that compared four treatments, \(\mathrm{A}, \mathrm{B}, \mathrm{C},\) and \(\mathrm{D},\) within each of three blocks, using a randomized block design. $$ \begin{array}{lrrrrrr} &&&{\text { Treatment }} \\ \hline \text { Block } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline 1 & 6 & 10 & 8 & 9 & 33 \\ 2 & 4 & 9 & 5 & 7 & 25 \\ 3 & 12 & 15 & 14 & 14 & 55 \\ \hline \text { Total } & 22 & 34 & 27 & 30 & 113 \end{array} $$ a. Do the data present sufficient evidence to indicate differences among the treatment means? Test using $$ \alpha=.05 . $$ b. Do the data present sufficient evidence to indicate differences among the block means? Test using \(\alpha=.05 .\) c. Rank the four treatment means using Tukey's method of paired comparisons with \(\alpha=.01\) d. Find a \(95 \%\) confidence interval for the difference in means for treatments \(\mathrm{A}\) and \(\mathrm{B}\). e. Does it appear that the use of a randomized block design for this experiment was justified? Explain.

The partially completed ANOVA table for a randomized block design is presented here: $$ \begin{array}{lcl} \text { Source } & d f & \text { SS } & \text { MS } \quad F \\ \hline \text { Treatments } & 4 & 14.2 & \\ \text { Blocks } & & 18.9 & \\ \text { Error } & 24 & & \\ \hline \text { Total } & 34 & 41.9 & \end{array} $$ a. How many blocks are involved in the design? b. How many observations are in each treatment total? c. How many observations are in each block total? d. Fill in the blanks in the ANOVA table. e. Do the data present sufficient evidence to indicate differences among the treatment means? Test using \(\alpha=.05\) f. Do the data present sufficient evidence to indicate differences among the block means? Test using \(\alpha=.05\)

In a study of starting salaries of assistant professors, \(^{8}\) five male assistant professors and five female assistant professors at each of three types of institutions granting doctoral degrees were polled and their initial starting salaries were recorded under the condition of anonymity. The results of the survey in \(\$ 1000\) are given in the following table. \begin{equation} \begin{array}{lccc} \text { Gender } & \text { Public Universities } & \text { Private/Independent } & \text { Church-Related } \\ \hline & \$ 57.3 & \$ 85.8 & \$ 78.9 \\ & 57.9 & 75.2 & 69.3 \\ \text { Males } & 56.5 & 66.9 & 69.7 \\ & 76.5 & 73.0 & 58.2 \\ & 62.0 & 73.0 & 61.2 \\ \hline & 47.4 & 62.1 & 60.4 \\ & 56.7 & 69.1 & 62.1 \\ \text { Females } & 69.0 & 66.5 & 59.8 \\ & 63.2 & 61.8 & 71.9 \\ & 65.3 & 76.7 & 61.6 \\ \hline \end{array} \end{equation} a. What type of design was used in collecting these data? b. Use an analysis of variance to test if there are significant differences in gender, in type of institution, and to test for a significant interaction of gender \(\times\) type of institution. c. Find a \(95 \%\) confidence interval estimate for the difference in starting salaries for male assistant professors and female assistant professors. Interpret this interval in terms of a gender difference in starting salaries. d. Use Tukey's procedure to investigate differences in assistant professor salaries for the three types of institutions. Use \(\alpha=.01\) e. Summarize the results of your analysis.