\begin{enumerate}
  \item David carries out an experiment with 4 identical dice, each with faces numbered 1 to 6 . He rolls the 4 dice and counts the number of dice showing an even number on the uppermost face. He repeats this 150 times. The results are summarised in the table below.
\end{enumerate}

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
No. of dice showing an even number & 0 & 1 & 2 & 3 & 4 \\
\hline
Frequency & 12 & 45 & 36 & 39 & 18 \\
\hline
\end{tabular}
\end{center}

David defines the random variable $C$ as the number of dice showing an even number on the uppermost face when the four dice are thrown.

David claims that $C \sim \mathrm {~B} ( 4,0.5 )$\\
(a) Stating your hypotheses clearly and using a $1 \%$ level of significance, test David's claim. Show your working clearly.

John claims that $C \sim \mathrm {~B} ( 4 , p )$\\
(b) Calculate an estimate of the value of $p$ from the summary of the results of David's experiment. Show your working clearly.

John decides to test his claim. He calculates expected frequencies using the results of David's experiment and obtains the following table.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
No. of dice showing an even number & 0 & 1 & 2 & 3 & 4 \\
\hline
Expected frequency & 8.65 & 36.00 & $d$ & 39.00 & $e$ \\
\hline
\end{tabular}
\end{center}

(c) Calculate, to 2 decimal places, the value of $d$ and the value of $e$\\
(d) State suitable hypotheses to test John's claim.

John obtained a test statistic of 16.9 and carries out a test at the $1 \%$ level of significance.\\
(e) State what conclusion John should make about his claim.

\begin{center}
\begin{tabular}{|l|l|}
\hline

\hline
END &  \\
\hline
\end{tabular}
\end{center}

\hfill \mbox{\textit{Edexcel S3 2018 Q6 [18]}}