| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Type I/II errors and power |
| Difficulty | Standard +0.3 This is a straightforward one-sample z-test application with known variance. Students must calculate the sample mean (21800/20 = 1090), perform a standard hypothesis test with H₀: μ = 1150 vs H₁: μ < 1150, and compute z = (1090-1150)/(105/√20) ≈ -2.55. The mechanics are routine for S2, though it requires careful statement of assumptions (independence of weeks) and correct interpretation of Type I error. Slightly easier than average due to clear setup and standard procedure. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks |
|---|---|
| (i) Assume pop sd same (105) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_1\): Pop mean < 1150 | B1 | Allow "\(\mu\)" but not just "mean" |
| \(\frac{21800/20 - 1150}{105/\sqrt{20}}\) | M1 | Allow \(\div \frac{105}{20}\). (Accept "totals" method) |
| \(= \pm 2.556\) or \(2.56\) | A1 | Or \(0.0053\) if prob/area comparison used |
| Compare with \(z = \pm 2.326\) (for a clear 2 tail test compare with \(\pm 2.576\)) | M1 | Correct comparison of \(z\) or prob/area consistent with their test |
| Evidence that mean distance decreased | A1 ft | [6] In context. Allow mean dist decreased If their \(z\) and/or clear 2 tail test |
| (ii) \(0.01\) | B1 | |
| Concluding there has been a decrease when there has not. | B1 | [2] In context |
**(i)** Assume pop sd same (105) | B1 |
$H_0$: Pop mean = 1150
$H_1$: Pop mean < 1150 | B1 | Allow "$\mu$" but not just "mean"
$\frac{21800/20 - 1150}{105/\sqrt{20}}$ | M1 | Allow $\div \frac{105}{20}$. (Accept "totals" method)
$= \pm 2.556$ or $2.56$ | A1 | Or $0.0053$ if prob/area comparison used
Compare with $z = \pm 2.326$ (for a clear 2 tail test compare with $\pm 2.576$) | M1 | Correct comparison of $z$ or prob/area consistent with their test
Evidence that mean distance decreased | A1 ft | [6] In context. Allow mean dist decreased If their $z$ and/or clear 2 tail test
**(ii)** $0.01$ | B1 |
Concluding there has been a decrease when there has not. | B1 | [2] In contextRecords show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km. New driving regulations are introduced and in the next 20 weeks he drives a total of 21 800 km.
\begin{enumerate}[label=(\roman*)]
\item Stating any assumption(s), test, at the 1% significance level, whether his mean weekly driving distance has decreased. [6]
\item A similar test at the 1% significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2011 Q5 [8]}}