A balanced experimental design has a sample size of \(n=11\) observations at each of \(k=3\) factor levels. The sample averages are \(\overline{x}_{1⋅}=48.05,\overline{x}_{2⋅}=44.74, and\ \overline{x}_{3⋅}=49.11, and\ MSE=4.96\).
(a) Calculate pairwise confidence intervals for the factor level means with an overall confidence level of 95%.
(b) Make a diagram showing which factor level means are known to be different and which ones are indistinguishable.
(c) What additional sampling would you recommend to reduce the lengths of the pairwise confidence intervals to no more than 2.0 ?

The analysis of variance table for a \(3 \times 4\) factorial experiment, with factor A at three levels and factor B at four levels, and with two observations per treatment is shown here:
Source, df, SS, MS, F
A, 2, 5.3
B, 3, 9.1
AB, 6,
Error, 12, 24.5
Total, 23, 43.7
a. Fill in the missing items in the table.
b. Do the data provide sufficient evidence to indicate that factors A and B interact? Test using \(\alpha \alpha=.05\). What are the practical implications of your answer?
c. Do the data provide sufficient evidence to indicate that factors A and B affect the response variable x? Explain.

Derive \(100(1-\alpha)\) percent lower and upper confidence intervals for p, when the data consist of the values of n independent Bernoulli random variables with parameter p.

An analysis of variance of the yields of five different varieties of wheat, observed on one plot each at each of six different locations. The data from this randomized block design are listed here:
\(\begin{array}{|c|c|} \hline Varieties & Location 1& Location 2& Location 3& Location 4& Location 5& Location 6 \\ \hline A& 35.3& 31.0& 32.7& 36.8& 37.2& 33.1 \\ \hline B& 30.7& 32.2& 31.4& 31.7& 35.0& 32.7\\ \hline C& 38.2& 33.4& 33.6& 37.1& 37.3& 38.2\\ \hline D& 34.9& 36.1& 35.2& 38.3& 40.2& 36.0\\ \hline E& 32.4& 28.9& 29.2& 30.7& 33.9& 32.1\\ \hline \end{array}\)
a. Use the appropriate nonparametric test to determine whether the data provide sufficient evidence to indicate a difference in the yields for the five different varieties of wheat. Test using \(\alpha\alpha=.05\).
b. How do the analysis of variance F test compare with the test in part a? Explain.

Assuming the null hypothesis is true, what is the probability that our z-test statistic would fall outside the z-critical boundaries (in the tails of the distribution the region of rejection) for anв \(\alpha=0.05?\)

\(\frac{8!}{(5!\times3!)}\)