| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2010 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Standard +0.3 This is a straightforward application of chi-squared tests with clearly structured data. Part (i) is a standard test of independence requiring calculation of expected frequencies and the test statistic, while part (ii) is a goodness-of-fit test. Both are routine procedures covered extensively in S3 with no conceptual challenges beyond following the standard algorithm. |
| Spec | 5.06a Chi-squared: contingency tables |
| Answer | Marks |
|---|---|
| B1 | For both hypotheses |
| M1, A1 | At least one correct |
| M1, A1, A1 | All correct |
| Correct form of any one | |
| All correct | |
| ART 9.64 | |
| M1, A1 | OR: \(P(z \geq 9.641) = 0.00806 < 0.05\) |
| 8 |
| Answer | Marks |
|---|---|
| M1, A1 | |
| M1, A1 | |
| M1 | OR:\(P(\chi^2 \geq 4.2) = 0.122 > 0.10\) |
| A1 | 6 |
| AEF in context |
## 7(i)
$H_0:$ Vegetable preference is independent of gender
$H_1:$ All alternatives
E-Values: 26 16.25 22.75
22 13.75 19.25
$\chi^2 = 5^2(26 + 22^2)/26 + 7.25^2(16.25^2 + 13.75^2)/... + 2.25^2(22.75^2 + 19.25^2)$
$= 9.641$
$9.64 > 5.991$
Reject $H_0$ (there is sufficient evidence at the 5% significance level that vegetable preference and gender are not independent)
| B1 | For both hypotheses |
| --- | --- |
| M1, A1 | At least one correct |
| M1, A1, A1 | All correct |
| | Correct form of any one |
| | All correct |
| | ART 9.64 |
| M1, A1 | OR: $P(z \geq 9.641) = 0.00806 < 0.05$ |
| | **8** |
## 7(ii)
($H_0:$ Vegetables have equal preference
$H_1:$ All alternatives)
Combining rows: 48 30 42
E-Values: 40 40 40
$\chi^2 = (8^2 + 10^2 + 2^2)/40 = 4.2$
$4.2 < 4.605$
Do not reject $H_0$, there is insufficient evidence at the 10% significance level of a difference in the proportion of preferred vegetables
| M1, A1 | |
| --- | --- |
| M1, A1 | |
| M1 | OR:$P(\chi^2 \geq 4.2) = 0.122 > 0.10$ |
| A1 | **6** |
| | AEF in context |
**Total: [14]**7 A chef wished to ascertain her customers' preference for certain vegetables. She asked a random sample of 120 customers for their preferred vegetable from asparagus, broad beans and cauliflower. The responses, classified according to the gender of the customer, are shown in the table.
(i) Test, at the $5 \%$ significance level, whether vegetable preference and gender are independent.\\
(ii) Determine whether, at the $10 \%$ significance level, the vegetables are equally preferred.
\hfill \mbox{\textit{OCR S3 2010 Q7 [14]}}