Three different medical procedures \((\mathrm{A}, \mathrm{B}\), and
\(\mathrm{C})\) for a certain disease are under investigation. For the study, \(3
\mathrm{~m}\) patients having this disease are to be selected and \(m\) are to be
assigned to each procedure. This common sample size \(m\) must be determined.
Let \(\mu_{1}, \mu_{2}\), and \(\mu_{3}\), be the means of the response of
interest under treatments A, B, and C, respectively. The hypotheses are:
\(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}\) versus \(H_{1}: \mu_{j} \neq \mu_{j^{\prime}}\)
for some \(j \neq j^{\prime} .\) To determine \(m\), from a pilot study the
experimenters use a guess of 30 of \(\sigma^{2}\) and they select the
significance level of \(0.05 .\) They are interested in detecting the pattern of
means: \(\mu_{2}=\mu_{1}+5\) and \(\mu_{3}=\mu_{1}+10\).
(a) Determine the noncentrality parameter under the above pattern of means.
(b) Use the \(\mathrm{R}\) function pf to determine the powers of the \(F\) -test
to detect the above pattern of means for \(m=5\) and \(m=10\).
(c) Determine the smallest value of \(m\) so that the power of detection is at
least \(0.80\)
(d) Answer (a)-(c) if \(\sigma^{2}=40\).
