| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | March |
| Marks | 10 |
| Topic | Chi-squared test of independence |
| Type | Interpret association after test |
| Difficulty | Standard +0.3 This is a straightforward chi-squared test of independence with a 2×3 contingency table. Students need to calculate expected frequencies, compute the test statistic, compare to critical value, and identify the largest contribution. While it requires multiple steps, it's a standard textbook application with no novel insight required, making it slightly easier than average for Further Maths statistics. |
| Spec | 5.06a Chi-squared: contingency tables |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Results | |||
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Win | Draw | Lose | |
| \multirow{2}{*}{Weather} | Sunny | 12 | 3 | 5 |
| \cline { 2 - 5 } | Not sunny | 8 | 12 | 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Expected frequencies | M1 | Find expected frequencies |
| A1 | ||
| 8 | 6 | 6 |
| 12 | 9 | 9 |
| M1 | Use \(\Sigma(O - E)^2/E\) | |
| \(\chi^2 = 2 + 1.5 + 0.167 + 1.333 + 1 + 0.111\) | A1 | Awrt 6.11 |
| \(= 6.111\) | A1 | Compare with 5.991, needs M1M1 |
| \(6.111 > 5.991\) | M1ft | ft on 6.111 only |
| Reject \(H_0\). Significance evidence of association between results and weather | A1ft | Contextualised, not too definite |
| Answer | Marks | Guidance |
|---|---|---|
| Sunny/Win | B1 | FT on \(\chi^2\) if M1M1 and evidence |
| Answer | Marks | Guidance |
|---|---|---|
| The team wins more than average in sunny conditions | B1 | Or equivalent |
## Part (i)
$H_0$: no association between weather and results, $H_1$: association
Expected frequencies | M1 | Find expected frequencies
| A1 |
| 8 | 6 | 6 |
|---|---|---|
| 12 | 9 | 9 |
| M1 | Use $\Sigma(O - E)^2/E$
$\chi^2 = 2 + 1.5 + 0.167 + 1.333 + 1 + 0.111$ | A1 | Awrt 6.11
$= 6.111$ | A1 | Compare with 5.991, needs M1M1
$6.111 > 5.991$ | M1ft | ft on 6.111 only
Reject $H_0$. Significance evidence of association between results and weather | A1ft | Contextualised, not too definite
## Part (ii)(a)
Sunny/Win | B1 | FT on $\chi^2$ if M1M1 and evidence
## Part (ii)(b)
The team wins more than average in sunny conditions | B1 | Or equivalent | FT from part (a)6 The captain of a sports team analyses the team's results according to the weather conditions, classified as "sunny" and "not sunny". The frequencies are shown in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{c|}{Results} \\
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & Win & Draw & Lose \\
\hline
\multirow{2}{*}{Weather} & Sunny & 12 & 3 & 5 \\
\cline { 2 - 5 }
& Not sunny & 8 & 12 & 10 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Test at the $5 \%$ significance level whether the team's performances are associated with weather conditions.
\item (a) Identify the cell that gives the largest contribution to the test statistic.\\
(b) Interpret your answer to part (ii)(a).
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q6 [10]}}