6 A scientist is investigating whether the ability to remember depends on age. A random sample of 150 students in different age groups is chosen. Each student is shown a set of 20 objects for thirty seconds and then asked to list as many as they can remember. The students are graded $A$ or $B$ according to how many objects they remembered correctly: grade $A$ for 16 or more correct and grade $B$ for fewer than 16 correct. The results are shown in the table.

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & \multicolumn{3}{c|}{Age of students} \\
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & $11 - 12$ years & $13 - 14$ years & $15 - 16$ years \\
\hline
Grade $A$ & 25 & 16 & 19 \\
\hline
Grade $B$ & 28 & 45 & 17 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Carry out a $\chi ^ { 2 }$-test at the $2.5 \%$ significance level to test whether grade is independent of age of student.\\

The scientist decides instead to use three grades: grade $A$ for 16 or more correct, grade $B$ for 10 to 15 correct and grade $C$ for fewer than 10 correct. The results are shown in the following table.

\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\multirow{2}{*}{} & \multicolumn{3}{|c|}{Age of students} \\
\hline
 & 11-12 years & 13-14 years & 15-16 years \\
\hline
Grade $A$ & 25 & 16 & 19 \\
\hline
Grade $B$ & 12 & 27 & 11 \\
\hline
Grade $C$ & 16 & 18 & 6 \\
\hline
\end{tabular}
\end{center}

With this second set of data, the test statistic is calculated as 10.91.
\item Complete the $\chi ^ { 2 }$-test at the $2.5 \%$ significance level for this second set of data.
\item State, with a reason, whether you would prefer to use the result from part (a) or part (b) to investigate whether the ability to remember depends on age.\\

If you use the following page to complete the answer to any question, the question number must be clearly shown.
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q6 [10]}}