A bowling alley manager in the UK is concerned about falling revenues. He collects data from the United States, hoping to use what he finds to revive his business in the UK.

He finds data which seem to show correlation between margarine consumption and bowling alley revenue. He attempts to carry out some statistical analysis in order to present his findings to the board of directors. He produces the scatter diagram shown below.

\includegraphics{figure_5}

The product moment correlation coefficient for these data is $-0.7617$. He carries out a one-tailed test at the 1\% level of significance and concludes that higher margarine consumption is associated with lower revenue generated by bowling alleys.

\begin{enumerate}[label=(\alph*)]
\item Show all the working for this test. [5]
\end{enumerate}

The manager also conducts a significance test for bowling alley revenue and fish consumption per person. He produces the computer output, shown below, for the analysis of bowling alley revenue versus fish consumption per person.

\# Pearson's product-moment correlation\\
\# data: revenue and fish\\
\# t = 3.8303, df = 8, p-value = 0.005215\\
\# alternative hypothesis: true correlation is not equal to 0\\
\# sample estimates:\\
\# correlation\\
\# 0.802423

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Comment on the correlation between bowling alley revenue and fish consumption per person and what the board of directors should do in light of the manager's findings in part (a) and part (b). [3]

\item Give one possible reason why the board of directors might not be happy with the manager's analysis. [1]
\end{enumerate}

\hfill \mbox{\textit{WJEC Unit 4 2019 Q5 [9]}}