A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that 40\% of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

\begin{enumerate}[label=(\alph*)]
\item Define the population associated with the magazine. [1]
\item Suggest a suitable sampling frame for the survey. [1]
\item Identify the sampling units. [1]
\item Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. [2]
\end{enumerate}

As a pilot study the editor took a random sample of 25 subscribers.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. [3]
\end{enumerate}

In fact only 6 subscribers agreed to the name being changed.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{5}
\item Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the percentage agreeing to the change is less that the editor believes. [5]
\end{enumerate}

The full survey is to be carried out using 200 randomly chosen subscribers.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{6}
\item Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. [7]
\end{enumerate}

\hfill \mbox{\textit{Edexcel S2  Q6 [20]}}