7 A dairy industry researcher, Robyn, decided to investigate the milk yield, classified as low, medium or high, obtained from four different breeds of cow, $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D .

The milk yield of a sample of 105 cows was monitored and the results are summarised in contingency Table 1.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{Table 1}} & \multicolumn{4}{|c|}{Yield} \\
\hline
 &  & Low & Medium & High & Total \\
\hline
\multirow{4}{*}{Breed} & A & 4 & 5 & 12 & 21 \\
\hline
 & B & 10 & 6 & 4 & 20 \\
\hline
 & C & 8 & 17 & 7 & 32 \\
\hline
 & D & 5 & 20 & 7 & 32 \\
\hline
 & Total & 27 & 48 & 30 & 105 \\
\hline
\end{tabular}
\end{center}

The sample of cows may be regarded as random.\\
Robyn decides to carry out a $\chi ^ { 2 }$-test for association between milk yield and breed using the information given in Table 1.

7
\begin{enumerate}[label=(\alph*)]
\item Contingency Table 2 gives some of the expected frequencies for this test.\\
Complete Table 2 with the missing expected values.

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{\multirow[t]{2}{*}{Table 2}} & \multicolumn{3}{|c|}{Yield} \\
\hline
 &  & Low & Medium & High \\
\hline
\multirow{4}{*}{Breed} & A &  &  & 6 \\
\hline
 & B & 5.14 & 9.14 & 5.71 \\
\hline
 & C &  &  &  \\
\hline
 & D & 8.23 & 14.63 & 9.14 \\
\hline
\end{tabular}
\end{center}

7
\item (i) For Robyn's test, the test statistic $\sum \frac { ( O - E ) ^ { 2 } } { E } = 19.4$ correct to three significant figures.\\
Use this information to carry out Robyn's test, using the $1 \%$ level of significance.\\

7 (b) (ii) By considering the observed frequencies given in Table 1 with the expected frequencies in Table 2, interpret, in context, the association, if any, between milk yield and breed.
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics  Q7 [9]}}