Standard +0.3 This is a standard S4 hypothesis testing question covering Type I/II errors and power calculations with binomial distributions. While it requires understanding of multiple statistical concepts and careful probability calculations using tables, the structure is routine for this module with clearly signposted parts (a)-(e) following a familiar pattern. The calculations are straightforward applications of binomial probability formulas rather than requiring novel problem-solving insight.
A drug is claimed to produce a cure to a certain disease in \(35 \%\) of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.
Write down suitable hypotheses to carry out this test.
Find the probability of making a Type I error.
The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of \(p\) where \(p\) is the probability of the drug curing a person with the disease.
P (cure)
0.2
0.3
0.4
0.5
P (Type II error)
0.5880
\(r\)
0.8565
\(s\)
Calculate the value of \(r\) and the value of \(s\).
Calculate the power of the test for \(p = 0.2\) and \(p = 0.4\)
Comment, giving your reasons, on the suitability of this test procedure.
\begin{enumerate}
\item A drug is claimed to produce a cure to a certain disease in $35 \%$ of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.\\
(a) Write down suitable hypotheses to carry out this test.\\
(b) Find the probability of making a Type I error.
\end{enumerate}
The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of $p$ where $p$ is the probability of the drug curing a person with the disease.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
P (cure) & 0.2 & 0.3 & 0.4 & 0.5 \\
\hline
P (Type II error) & 0.5880 & $r$ & 0.8565 & $s$ \\
\hline
\end{tabular}
\end{center}
(c) Calculate the value of $r$ and the value of $s$.\\
(d) Calculate the power of the test for $p = 0.2$ and $p = 0.4$\\
(e) Comment, giving your reasons, on the suitability of this test procedure.
\hfill \mbox{\textit{Edexcel S4 2008 Q6 [12]}}