| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Hypothesis test for mean |
| Difficulty | Standard +0.3 This is a straightforward one-sample z-test for a mean using the CLT with n=75. Students must state the assumption (σ unchanged), calculate a test statistic, compare to critical value, and explain why CLT applies (large sample). All steps are routine applications of standard hypothesis testing procedure with no novel insight required, making it slightly easier than average. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Assume sd unchanged or \(sd = 10.4\) | B1 | Or e.g. var unchanged |
| \(H_0\): Pop mean speed (or \(\mu\)) \(= 62.3\); \(H_1\): Pop mean speed (or \(\mu\)) \(< 62.3\) | B1 | Both. Not just "Mean..." |
| \(\frac{59.9 - 62.3}{\frac{10.4}{\sqrt{75}}}\) | M1 | Accept sd/var mixes, but must have \(\sqrt{75}\) |
| \(= -1.999\) or \(-2.00\) (allow \(+\) or \(-\)) | A1 | Correct \(z\) value (or correct critical value) |
| Compare \(-2.054\) or \(-2.055\) | M1 | Valid comparison of \(z\)'s/areas/critical values |
| No evidence that mean speed decreased | A1ft | [6] No contradictions. Do not ft 2-tail test. |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Pop distribution unknown | B1 | |
| Yes | B1 | [2] |
## Question 2(i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Assume sd unchanged or $sd = 10.4$ | B1 | Or e.g. var unchanged |
| $H_0$: Pop mean speed (or $\mu$) $= 62.3$; $H_1$: Pop mean speed (or $\mu$) $< 62.3$ | B1 | Both. Not just "Mean..." |
| $\frac{59.9 - 62.3}{\frac{10.4}{\sqrt{75}}}$ | M1 | Accept sd/var mixes, but must have $\sqrt{75}$ |
| $= -1.999$ or $-2.00$ (allow $+$ or $-$) | A1 | Correct $z$ value (or correct critical value) |
| Compare $-2.054$ or $-2.055$ | M1 | Valid comparison of $z$'s/areas/critical values |
| No evidence that mean speed decreased | A1ft | **[6]** No contradictions. Do not ft 2-tail test. |
## Question 2(ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Pop distribution unknown | B1 | |
| Yes | B1 | **[2]** |
---2 A traffic officer notes the speeds of vehicles as they pass a certain point. In the past the mean of these speeds has been $62.3 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and the standard deviation has been $10.4 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. A speed limit is introduced, and following this, the mean of the speeds of 75 randomly chosen vehicles passing the point is found to be $59.9 \mathrm {~km} \mathrm {~h} ^ { - 1 }$.\\
(i) Making an assumption that should be stated, test at the $2 \%$ significance level whether the mean speed has decreased since the introduction of the speed limit.\\
(ii) Explain whether it was necessary to use the Central Limit theorem in part (i).
\hfill \mbox{\textit{CAIE S2 2013 Q2 [8]}}