| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Two-tail z-test |
| Difficulty | Moderate -0.3 This is a straightforward one-sample z-test with known variance following a standard template: calculate test statistic, compare to critical value, and state conclusion. Part (b) requires recall of Type II error definition. The question is slightly easier than average because all values are given directly, the calculation is routine, and no conceptual complications arise—typical textbook S2 material. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: \mu = 35\), \(H_1: \mu \neq 35\) | B1 | |
| 2-tail test, 1% significance level | ||
| Under \(H_0\), \(\bar{X} \sim N\!\left(35, \frac{\sigma^2}{n}\right)\) | ||
| \(\bar{X} \sim N\!\left(35, \frac{144}{100}\right)\) | B1 | |
| \(z = \frac{37.9 - 35}{1.2}\) | M1 | \(z = \frac{37.9-35}{\text{their } \sigma/\sqrt{n}}\) |
| \(z = 2.42\) | A1\(\checkmark\) | On their \(\sigma/\sqrt{n}\) |
| \(z_{crit} = \pm 2.5758\) | B1 | |
| Do not reject \(H_0\) | A1\(\checkmark\) | On their \(z\) |
| Evidence to support the claim that the mean age is 35 years. | E1\(\checkmark\) | |
| Total: 7 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Accept \(H_0\) when \(H_0\) false. Accepting the mean to be 35 years when it isn't. | B2 | Allow B1 if not in context |
| Total: 2 marks |
# Question 8:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 35$, $H_1: \mu \neq 35$ | B1 | |
| 2-tail test, 1% significance level | | |
| Under $H_0$, $\bar{X} \sim N\!\left(35, \frac{\sigma^2}{n}\right)$ | | |
| $\bar{X} \sim N\!\left(35, \frac{144}{100}\right)$ | B1 | |
| $z = \frac{37.9 - 35}{1.2}$ | M1 | $z = \frac{37.9-35}{\text{their } \sigma/\sqrt{n}}$ |
| $z = 2.42$ | A1$\checkmark$ | On their $\sigma/\sqrt{n}$ |
| $z_{crit} = \pm 2.5758$ | B1 | |
| Do not reject $H_0$ | A1$\checkmark$ | On their $z$ |
| Evidence to support the claim that the mean age is 35 years. | E1$\checkmark$ | |
| **Total: 7 marks** | | |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Accept $H_0$ when $H_0$ false. Accepting the mean to be 35 years when it isn't. | B2 | Allow B1 if not in context |
| **Total: 2 marks** | | |8 The mean age of people attending a large concert is claimed to be 35 years.
A random sample of 100 people attending the concert was taken and their mean age was found to be 37.9 years.
\begin{enumerate}[label=(\alph*)]
\item Given that the standard deviation of the ages of the people attending the concert is 12 years, test, at the $1 \%$ level of significance, the claim that the mean age is 35 years.\\
(7 marks)
\item Explain, in the context of this question, the meaning of a Type II error.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2005 Q8 [9]}}