4. A quality control manager regularly samples 20 items from a production line and records the number of defective items $x$. The results of 100 such samples are given in table 1 below.

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 or more \\
\hline
Frequency & 17 & 31 & 19 & 14 & 9 & 7 & 3 & 0 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Estimate the proportion of defective items from the production line.

The manager claimed that the number of defective items in a sample of 20 can be modelled by a binomial distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2.

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 or more \\
\hline
\begin{tabular}{ c }
Expected \\
frequency \\
\end{tabular} & 12.2 & 27.0 & $r$ & 19.0 & $s$ & 3.2 & 0.9 & 0.2 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{center}
\end{table}
\item Find the value of $r$ and the value of $s$ giving your answers to 1 decimal place.
\item Stating your hypotheses clearly, use a $5 \%$ level of significance to test the manager's claim.
\item Explain what the analysis in part (c) tells the manager about the occurrence of defective items from this production line.
\end{enumerate}

\hfill \mbox{\textit{Edexcel S3 2007 Q4 [13]}}