| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (lower tail) |
| Difficulty | Standard +0.3 This is a straightforward application of the z-test formula requiring algebraic rearrangement to find n, followed by a standard hypothesis test procedure. While it involves multiple steps, each step uses routine techniques (rearranging z = (x̄ - μ)/(s/√n), comparing to critical values) with no conceptual challenges beyond standard S2 content. |
| Spec | 2.05e Hypothesis test for normal mean: known variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(z = \frac{64.3 - 65}{4.9/\sqrt{n}} = -1.807\), \(n = 160\) | M1, M1, A1 (3 marks) | For standardising equation \(= +/-1.807\) with \(n\) or \(\sqrt{n}\). Solving for \(n\). For correct answer CWO. |
| (ii) \(H_0: \mu = 65\), \(H_1: \mu < 65\) | B1, B1 | For \(H_0\) and \(H_1\). |
| Critical Value \(+/-1.645\) | B1 | For \(+/-1.645\) (or \(\bar{x} +/- 1.96\) for two tail test). |
| Significant growth decrease | M1, A1 (4 marks) | Comparing given statistic with their CV. Correct conclusion. |
(i) $z = \frac{64.3 - 65}{4.9/\sqrt{n}} = -1.807$, $n = 160$ | M1, M1, A1 (3 marks) | For standardising equation $= +/-1.807$ with $n$ or $\sqrt{n}$. Solving for $n$. For correct answer CWO.
(ii) $H_0: \mu = 65$, $H_1: \mu < 65$ | B1, B1 | For $H_0$ and $H_1$.
Critical Value $+/-1.645$ | B1 | For $+/-1.645$ (or $\bar{x} +/- 1.96$ for two tail test).
Significant growth decrease | M1, A1 (4 marks) | Comparing given statistic with their CV. Correct conclusion.
---From previous years' observations, the lengths of salmon in a river were found to be normally distributed with mean 65 cm. A researcher suspects that pollution in water is restricting growth. To test this theory, she measures the length $x$ cm of a random sample of $n$ salmon and calculates that $\bar{x} = 64.3$ and $s = 4.9$, where $s^2$ is the unbiased estimate of the population variance. She then carries out an appropriate hypothesis test.
\begin{enumerate}[label=(\roman*)]
\item Her test statistic $z$ has a value of $-1.807$ correct to 3 decimal places. Calculate the value of $n$. [3]
\item Using this test statistic, carry out the hypothesis test at the 5% level of significance and state what her conclusion should be. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2002 Q3 [7]}}