| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Expected frequencies partially provided |
| Difficulty | Moderate -0.3 This is a straightforward chi-squared test of independence with standard structure: write hypotheses, calculate one expected frequency and verify a given contribution, then complete the test with provided test statistic. All calculations are routine with no conceptual challenges—slightly easier than average because most computational work is already done for the student. |
| Spec | 5.06a Chi-squared: contingency tables |
| Recycling Service | ||||||
| \cline { 3 - 5 } \multicolumn{2}{|c|}{} | Standard | Special | Both | |||
| Small | 35 | 26 | 44 | ||
| Large | 55 | 52 | 73 | |||
| Recycling Service | ||||||
| \cline { 3 - 5 } \multicolumn{2}{|c|}{} | Standard | Special | Both | |||
| Small | 0.1023 | 0.2607 | 0.0186 | ||
| Large | 0.0597 | 0.1520 | 0.0108 | |||
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): no association between size of business and recycling service used; \(H_1\): some association between size of business and recycling service used | B1 for both | [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Expected frequency \(= \frac{78}{285} \times 180 = 49.2632\); Contribution \(= \frac{(52-49.2632)^2}{49.2632} = 0.1520\) | M1 A1, M1 for valid attempt at \(\frac{(O-E)^2}{E}\), A1 NB Answer given; allow 0.152 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Test statistic \(X^2 = 0.6041\); refer to \(\chi^2_2\); critical value at 5% \(= 5.991\); result is not significant; no evidence of association between size of business and recycling service used | B1, B1 for 2 deg of freedom, B1 CAO for cv, B1 for not significant, E1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu = 32.8\); \(H_1: \mu < 32.8\) where \(\mu\) denotes population mean weight of rubbish; test statistic \(= \frac{30.9-32.8}{3.4/\sqrt{50}} = \frac{-1.9}{0.4808} = -3.951\); 5% one-tailed critical value \(z = -1.645\); \(-3.951 < -1.645\) so significant; sufficient evidence to reject \(H_0\); evidence that weight of rubbish in dustbins has been reduced | B1 for use of 32.8, B1 for both correct, B1 for definition of \(\mu\), M1 must include \(\sqrt{50}\), A1, B1 for \(\pm 1.645\), M1 for comparison, A1 for conclusion in context | [8] |
# Question 4:
## Part (i)
| $H_0$: no association between size of business and recycling service used; $H_1$: some association between size of business and recycling service used | B1 for both | [1] |
## Part (ii)
| Expected frequency $= \frac{78}{285} \times 180 = 49.2632$; Contribution $= \frac{(52-49.2632)^2}{49.2632} = 0.1520$ | M1 A1, M1 for valid attempt at $\frac{(O-E)^2}{E}$, A1 **NB Answer given**; allow 0.152 | [4] |
## Part (iii)
| Test statistic $X^2 = 0.6041$; refer to $\chi^2_2$; critical value at 5% $= 5.991$; result is not significant; no evidence of association between size of business and recycling service used | B1, B1 for 2 deg of freedom, B1 CAO for cv, B1 for not significant, E1 | [5] |
## Part (iv)
| $H_0: \mu = 32.8$; $H_1: \mu < 32.8$ where $\mu$ denotes population mean weight of rubbish; test statistic $= \frac{30.9-32.8}{3.4/\sqrt{50}} = \frac{-1.9}{0.4808} = -3.951$; 5% one-tailed critical value $z = -1.645$; $-3.951 < -1.645$ so significant; sufficient evidence to reject $H_0$; evidence that weight of rubbish in dustbins has been reduced | B1 for use of 32.8, B1 for both correct, B1 for definition of $\mu$, M1 must include $\sqrt{50}$, A1, B1 for $\pm 1.645$, M1 for comparison, A1 for conclusion in context | [8] |4 A council provides waste paper recycling services for local businesses. Some businesses use the standard service for recycling paper, others use a special service for dealing with confidential documents, and others use both. Businesses are classified as small or large. A survey of a random sample of 285 businesses gives the following data for size of business and recycling service.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
\multicolumn{2}{|c|}{} & \multicolumn{3}{|c|}{Recycling Service} \\
\cline { 3 - 5 }
\multicolumn{2}{|c|}{} & Standard & Special & Both \\
\hline
\begin{tabular}{ l }
Size of \\
business \\
\end{tabular} & Small & 35 & 26 & 44 \\
\hline
& Large & 55 & 52 & 73 \\
\hline
\end{tabular}
\end{center}
(i) Write down null and alternative hypotheses for a test to examine whether there is any association between size of business and recycling service used.
The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
\multicolumn{2}{|c|}{} & \multicolumn{3}{|c|}{Recycling Service} \\
\cline { 3 - 5 }
\multicolumn{2}{|c|}{} & Standard & Special & Both \\
\hline
\begin{tabular}{ l }
Size of \\
business \\
\end{tabular} & Small & 0.1023 & 0.2607 & 0.0186 \\
\hline
& Large & 0.0597 & 0.1520 & 0.0108 \\
\hline
\end{tabular}
\end{center}
The sum of these contributions is 0.6041 .\\
(ii) Calculate the expected frequency for large businesses using the special service. Verify the corresponding contribution 0.1520 to the test statistic.\\
(iii) Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly.
The council is also investigating the weight of rubbish in domestic dustbins. In 2008 the average weight of rubbish in bins was 32.8 kg . The council has now started a recycling initiative and wishes to determine whether there has been a reduction in the weight of rubbish in bins. A random sample of 50 domestic dustbins is selected and it is found that the mean weight of rubbish per bin is now 30.9 kg , and the standard deviation is 3.4 kg .\\
(iv) Carry out a test at the $5 \%$ level to investigate whether the mean weight of rubbish has been reduced in comparison with 2008 . State carefully your null and alternative hypotheses.
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\hfill \mbox{\textit{OCR MEI S2 2010 Q4 [18]}}