| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Interpret association after test |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with clearly presented contingency table data. Students follow a routine procedure: state hypotheses, calculate expected frequencies, compute chi-squared statistic using the formula, compare to critical value, and interpret. Part (ii) requires basic comparison of observed vs expected values. While it involves multiple calculations, it's a textbook application with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Sex | \multirow{2}{*}{Row totals} | ||
| Male | Female | |||
| \multirow{5}{*}{Type of car} | Hatchback | 96 | 36 | 132 |
| Saloon | 77 | 35 | 112 | |
| People carrier | 38 | 44 | 82 | |
| 4WD | 19 | 8 | 27 | |
| Sports car | 22 | 25 | 47 | |
| Column totals | 252 | 148 | 400 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): no association between type of car and sex; \(H_1\): some association between type of car and sex | B1 | |
| Expected values: Hatchback M: 83.16, F: 48.84; Saloon M: 70.56, F: 41.44; People carrier M: 51.66, F: 30.34; 4WD M: 17.01, F: 9.99; Sports car M: 29.61, F: 17.39 | M1, A2 | M1 A2 for expected values to 2dp (allow A1 for at least one row or column correct) |
| Contributions \((O-E)^2/E\): Hatchback M: 1.98, F: 3.38; Saloon M: 0.59, F: 1.00; People carrier M: 3.61, F: 6.15; 4WD M: 0.23, F: 0.40; Sports car M: 1.96, F: 3.33 | M1, A1 | M1 for valid attempt at \((O-E)^2/E\); A1 for all correct. NB These M1A1 marks cannot be implied by a correct final value of \(X^2\) |
| \(X^2 = 22.62\) | M1, A1 | M1 for summation; A1 for \(X^2\) CAO |
| Refer to \(\chi^2_4\); critical value at 5% level \(= 9.488\) | B1, B1 | B1 for 4 deg of f; B1 CAO for cv |
| \(22.62 > 9.488\); result is significant; there is evidence to suggest some association between sex and type of car | M1, A1 | M1 sensible comparison leading to conclusion; A1. NB if \(H_0\), \(H_1\) reversed, or 'correlation' mentioned, do not award first B1 or final A1. Total: 12 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| In hatchbacks, male drivers are more frequent than expected | E1 | |
| In saloons, male drivers are slightly more frequent than expected | E1 | |
| In people carriers, female drivers are much more frequent than expected | E1 | |
| In 4WDs the numbers are roughly as expected | E1 | |
| In sports cars, female drivers are more frequent than expected | E1 | Total: 5 |
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: no association between type of car and sex; $H_1$: some association between type of car and sex | B1 | |
| Expected values: Hatchback M: 83.16, F: 48.84; Saloon M: 70.56, F: 41.44; People carrier M: 51.66, F: 30.34; 4WD M: 17.01, F: 9.99; Sports car M: 29.61, F: 17.39 | M1, A2 | M1 A2 for expected values to 2dp (allow A1 for at least one row or column correct) |
| Contributions $(O-E)^2/E$: Hatchback M: 1.98, F: 3.38; Saloon M: 0.59, F: 1.00; People carrier M: 3.61, F: 6.15; 4WD M: 0.23, F: 0.40; Sports car M: 1.96, F: 3.33 | M1, A1 | M1 for valid attempt at $(O-E)^2/E$; A1 for all correct. NB These M1A1 marks cannot be implied by a correct final value of $X^2$ |
| $X^2 = 22.62$ | M1, A1 | M1 for summation; A1 for $X^2$ CAO |
| Refer to $\chi^2_4$; critical value at 5% level $= 9.488$ | B1, B1 | B1 for 4 deg of f; B1 CAO for cv |
| $22.62 > 9.488$; result is significant; there is evidence to suggest some association between sex and type of car | M1, A1 | M1 sensible comparison leading to conclusion; A1. NB if $H_0$, $H_1$ reversed, or 'correlation' mentioned, do not award first B1 or final A1. **Total: 12** |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| In hatchbacks, male drivers are more frequent than expected | E1 | |
| In saloons, male drivers are slightly more frequent than expected | E1 | |
| In people carriers, female drivers are much more frequent than expected | E1 | |
| In 4WDs the numbers are roughly as expected | E1 | |
| In sports cars, female drivers are more frequent than expected | E1 | **Total: 5** |4 In a traffic survey a random sample of 400 cars passing a particular location during the rush hour is selected. The type of car and the sex of the driver are classified as follows.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{2}{|c|}{Sex} & \multirow{2}{*}{Row totals} \\
\hline
& & Male & Female & \\
\hline
\multirow{5}{*}{Type of car} & Hatchback & 96 & 36 & 132 \\
\hline
& Saloon & 77 & 35 & 112 \\
\hline
& People carrier & 38 & 44 & 82 \\
\hline
& 4WD & 19 & 8 & 27 \\
\hline
& Sports car & 22 & 25 & 47 \\
\hline
\multicolumn{2}{|c|}{Column totals} & 252 & 148 & 400 \\
\hline
\end{tabular}
\end{center}
(i) Carry out a test at the $5 \%$ significance level to examine whether there is any association between type of car and sex of driver. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.\\
(ii) For each type of car, comment briefly on how the number of drivers of each sex compares with what would be expected if there were no association.
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\hfill \mbox{\textit{OCR MEI S2 2009 Q4 [17]}}