Get help with upper level probability

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 reading test scores of students at a rural Texas school district was carried out in Current Issues in Education (Jan. 2014). Students were classified as attending elementary, middle, or high school and whether they passed a reading comprehension test. Toe data for the sample of 1,012 students are summarized in the accompanying table. Does the passing rate on the reading comprehension test in Texas differ for elementary, middle, and high school students? Use \(\alpha=10\).
\(\begin{array}{|l|c|c|c|} \hline& \text {Elementary School} & \text {Middle School} & \text {High School} \\ \hline \text {Number Passing} &372 &418 & 143 \\ \hline \text {Number Failing} &44 &25 & 10 \\ \hline \text {Totals } &416 & 443 & 153\\ \hline \end{array} \)