| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2013 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Cell combining required |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with a straightforward complication (combining cells due to low expected frequencies). The calculation of expected values is routine, recognizing the need to combine cells is a textbook application of the 'expected frequency ≥ 5' rule, and executing the test follows a standard algorithm. Slightly above average difficulty due to the cell-combining requirement, but this is a common S3 exercise type. |
| Spec | 5.06a Chi-squared: contingency tables |
| Subject | ||||
| \cline { 2 - 5 } | Biology | Chemistry | Art | |
| \cline { 2 - 5 } Gender | Male | 13 | 4 | 11 |
| \cline { 2 - 5 } | Female | 37 | 8 | 7 |
| \cline { 2 - 5 } | ||||
| \cline { 2 - 5 } | ||||
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(17.5\quad 4.2\quad 6.3\) | M1 | eg \(50 \times 28 \div 80\) |
| \(32.5\quad 7.8\quad 11.7\) oe | A1 | At least 2 correct |
| A1 | All correct | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| The E value of \(4.2 < 5\) | B1 | Need not mention 4.2 |
| Combine Biology and Chemistry (both sciences) | B1 | May need to look at (iii) to see which subjects combined |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): Subject and sex are independent | ||
| \(H_1\): They are not independent | B1 | oe. NOT 'variables', 'they' etc |
| \(21.7\quad 6.3\) | B1 | or \(17.5\quad 10.5\) |
| \(40.3\quad 11.7\) | \(32.5\quad 19.5\) if C/A combined | |
| \(\chi^2 = (4.7-0.5)^2(21.7^{-1} + 6.3^{-1} + 40.3^{-1} + 11.7^{-1})\) | M1M1 | No Yates (inc \(\upsilon > 1\)) or incorrect Yates (eg no modulus) M1M0 |
| \(= 5.558\ldots\) | A1 | allow 6.96 or 6.79; Chem./Art combined B1B1M1M1A0B1M1A0 \((TS = 3.75)\) |
| \((\nu = 1)\) | ||
| \((\alpha)\ 2\tfrac{1}{2}\%\ \text{CV} = 5.024\) | B1 | |
| \(5.558 > \text{CV}\) or in CR and reject \(H_0\) | M1 | ft TS & CV. Correct first conclusion. If C/A prob. accept \(H_0\) |
| \((\beta)\ P(\chi^2_1 \geq 5.558) = 0.0184\) | B1 | |
| \(< 0.025\) and reject \(H_0\) | M1 | |
| There is significant evidence that subject and sex are not independent | A1 | cwo. NOT over-assertive. Thus no or incorrect Yates can score max 6/8; B1B1M1M0A1B1M1A0 |
| [8] |
# Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $17.5\quad 4.2\quad 6.3$ | M1 | eg $50 \times 28 \div 80$ |
| $32.5\quad 7.8\quad 11.7$ oe | A1 | At least 2 correct |
| | A1 | All correct |
| | **[3]** | |
---
# Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| The E value of $4.2 < 5$ | B1 | Need not mention 4.2 |
| Combine Biology and Chemistry (both sciences) | B1 | May need to look at (iii) to see which subjects combined |
| | **[2]** | |
---
# Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: Subject and sex are independent | | |
| $H_1$: They are not independent | B1 | oe. NOT 'variables', 'they' etc |
| $21.7\quad 6.3$ | B1 | or $17.5\quad 10.5$ |
| $40.3\quad 11.7$ | | $32.5\quad 19.5$ if C/A combined |
| $\chi^2 = (4.7-0.5)^2(21.7^{-1} + 6.3^{-1} + 40.3^{-1} + 11.7^{-1})$ | M1M1 | No Yates (inc $\upsilon > 1$) or incorrect Yates (eg no modulus) M1M0 |
| $= 5.558\ldots$ | A1 | allow 6.96 or 6.79; Chem./Art combined B1B1M1M1A0B1M1A0 $(TS = 3.75)$ |
| $(\nu = 1)$ | | |
| $(\alpha)\ 2\tfrac{1}{2}\%\ \text{CV} = 5.024$ | B1 | |
| $5.558 > \text{CV}$ or in CR and reject $H_0$ | M1 | ft TS & CV. Correct first conclusion. If C/A prob. accept $H_0$ |
| $(\beta)\ P(\chi^2_1 \geq 5.558) = 0.0184$ | B1 | |
| $< 0.025$ and reject $H_0$ | M1 | |
| There is significant evidence that subject and sex are not independent | A1 | cwo. NOT over-assertive. Thus no or incorrect Yates can score max 6/8; B1B1M1M0A1B1M1A0 |
| | **[8]** | |
---6 A random sample of 80 students who had all studied Biology, Chemistry and Art at a college was each asked which they enjoyed most. The results, classified according to gender, are given in the table.
\begin{center}
\begin{tabular}{ l | l | c | c | c | }
& \multicolumn{4}{c}{Subject} \\
\cline { 2 - 5 }
& & Biology & Chemistry & Art \\
\cline { 2 - 5 }
Gender & Male & 13 & 4 & 11 \\
\cline { 2 - 5 }
& Female & 37 & 8 & 7 \\
\cline { 2 - 5 }
& & & & \\
\cline { 2 - 5 }
\end{tabular}
\end{center}
It is required to carry out a test of independence between subject most enjoyed and gender at the $2 \frac { 1 } { 2 } \%$ significance level.\\
(i) Calculate the expected values for the cells.\\
(ii) Explain why it is necessary to combine cells, and choose a suitable combination.\\
(iii) Carry out the test.
\hfill \mbox{\textit{OCR S3 2013 Q6 [13]}}