| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 7 |
| Topic | Chi-squared test of independence |
| Type | Chi-squared test with algebraic entries |
| Difficulty | Standard +0.8 This question requires understanding of chi-squared test mechanics including deriving the O-E expression algebraically from a partially-specified contingency table (non-trivial algebra with row/column totals), then conducting a hypothesis test with correct degrees of freedom. The algebraic manipulation in part (a) is more demanding than standard chi-squared questions, elevating it above average difficulty. |
| Spec | 5.06a Chi-squared: contingency tables |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Main type of investment | |||
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Bonds | Cash | Stocks | |
| \multirow{2}{*}{Age} | \(25 - 44\) | \(a\) | \(b - e\) | \(e\) |
| \cline { 2 - 5 } | \(45 - 75\) | \(c\) | \(d - 59\) | 59 |
5. Charlie carried out a survey on the main type of investment people have.
The contingency table below shows the results of a survey of a random sample of people.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{c|}{Main type of investment} \\
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & Bonds & Cash & Stocks \\
\hline
\multirow{2}{*}{Age} & $25 - 44$ & $a$ & $b - e$ & $e$ \\
\cline { 2 - 5 }
& $45 - 75$ & $c$ & $d - 59$ & 59 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find an expression, in terms of $a , b , c$ and $d$, for the difference between the observed and the expected value $( O - E )$ for the group whose main type of investment is Bonds and are aged $45 - 75$\\
Express your answer as a single fraction in its simplest form.
Given that $\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.62$ for this information,
\item test, at the $5 \%$ level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2022 Q5 [7]}}