3 A random sample of 50 values of the continuous random variable $X$ was taken. These values are summarised in the following table.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Interval & $1 \leqslant x < 1.5$ & $1.5 \leqslant x < 2$ & $2 \leqslant x < 2.5$ & $2.5 \leqslant x < 3$ & $3 \leqslant x < 3.5$ & $3.5 \leqslant x \leqslant 4$ \\
\hline
Observed frequency & 3 & 3 & 8 & 11 & 13 & 12 \\
\hline
\end{tabular}
\end{center}

It is required to test the goodness of fit of the distribution with probability density function $f$ given by

$$f ( x ) = \begin{cases} \frac { 1 } { 24 } \left( \frac { 4 } { x ^ { 2 } } + x ^ { 2 } \right) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

The expected frequencies, correct to 4 decimal places, are given in the following table.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Interval & $1 \leqslant x < 1.5$ & $1.5 \leqslant x < 2$ & $2 \leqslant x < 2.5$ & $2.5 \leqslant x < 3$ & $3 \leqslant x < 3.5$ & $3.5 \leqslant x \leqslant 4$ \\
\hline
Expected frequency & 4.4271 & $a$ & 6.1285 & 8.4549 & $b$ & 14.9678 \\
\hline
\end{tabular}
\end{center}

(a) Show that $a = 4.6007$ and find the value of $b$.\\

(b) Carry out a goodness of fit test, at the $10 \%$ significance level, to test whether f is a satisfactory model for the data.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q3}}