4
\begin{enumerate}[label=(\alph*)]
\item An amateur weather forecaster has been keeping records of air pressure, measured in atmospheres. She takes the measurement at the same time every day using a barometer situated in her garden. A random sample of 100 of her observations is summarised in the table below. The corresponding expected frequencies for a Normal distribution, with its two parameters estimated by sample statistics, are also shown in the table.

\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Pressure ( $a$ atmospheres) & Observed frequency & Frequency as given by Normal model \\
\hline
$a \leqslant 0.98$ & 4 & 1.45 \\
\hline
$0.98 < a \leqslant 0.99$ & 6 & 5.23 \\
\hline
$0.99 < a \leqslant 1.00$ & 9 & 13.98 \\
\hline
$1.00 < a \leqslant 1.01$ & 15 & 23.91 \\
\hline
$1.01 < a \leqslant 1.02$ & 37 & 26.15 \\
\hline
$1.02 < a \leqslant 1.03$ & 21 & 18.29 \\
\hline
$1.03 < a$ & 8 & 10.99 \\
\hline
\end{tabular}
\end{center}

Carry out a test at the $5 \%$ level of significance of the goodness of fit of the Normal model. State your conclusion carefully and comment on your findings.
\item The forecaster buys a new digital barometer that can be linked to her computer for easier recording of observations. She decides that she wishes to compare the readings of the new barometer with those of the old one. For a random sample of 10 days, the readings (in atmospheres) of the two barometers are shown below.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Day & A & B & C & D & E & F & G & H & I & J \\
\hline
Old & 0.992 & 1.005 & 1.001 & 1.011 & 1.026 & 0.980 & 1.020 & 1.025 & 1.042 & 1.009 \\
\hline
New & 0.985 & 1.003 & 1.002 & 1.014 & 1.022 & 0.988 & 1.030 & 1.016 & 1.047 & 1.025 \\
\hline
\end{tabular}
\end{center}

Use an appropriate Wilcoxon test to examine at the $10 \%$ level of significance whether there is any reason to suppose that, on the whole, readings on the old and new barometers do not agree.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI S3 2007 Q4 [18]}}