| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (upper tail) |
| Difficulty | Moderate -0.3 This is a straightforward one-sample z-test with known variance following a standard template: state hypotheses, calculate test statistic using given formula, compare to critical value, and conclude. Part (b) tests basic understanding of Type I/II errors through direct application of definitions. While it requires multiple steps, each is routine for S2 students with no novel problem-solving or insight needed. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu = 4.0\) | B1 | (both) |
| \(H_1: \mu > 4.0\) | ||
| \(z_{calc} = \frac{4.2 - 4}{1.1/\sqrt{40}} = 1.15\) | M1 | Alternative: \(\mathbb{P}(\bar{X} > 4.2) = \mathbb{P}(Z > 1.15)\) M1A1 |
| A1 | awrt | |
| \(z_{crit} = 1.6449\) | B1 | \(= 1 - 0.87493 = 0.125\) B1 |
| \(0.125 > 0.05 \Rightarrow\) accept \(H_0\) Adep1 | ||
| Accept \(H_0\) [or Reject \(H_1\)] | A1 | Dep on B1M1B1 |
| Insufficient evidence at 5% level to support Julian's claim | E1 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Type II error. Accepted \(H_0\) when \(H_0\) was false (oe) | B1ft | Follow through on conclusion in (a) |
| E1 | 2 | Dep on previous mark |
| If Reject \(H_0\) in (a) then: No error (B1fi) Rejected \(H_0\) when \(H_0\) was false (oe) (E1) | ||
| 8 |
### 2(a)
| $H_0: \mu = 4.0$ | B1 | (both) |
| --- | --- | --- |
| $H_1: \mu > 4.0$ | | |
| $z_{calc} = \frac{4.2 - 4}{1.1/\sqrt{40}} = 1.15$ | M1 | **Alternative:** $\mathbb{P}(\bar{X} > 4.2) = \mathbb{P}(Z > 1.15)$ M1A1 |
| | A1 | awrt |
| $z_{crit} = 1.6449$ | B1 | $= 1 - 0.87493 = 0.125$ **B1** |
| | | $0.125 > 0.05 \Rightarrow$ accept $H_0$ **Adep1** |
| **Accept $H_0$ [or Reject $H_1$]** | A1 | Dep on B1M1B1 |
| **Insufficient evidence at 5% level to support Julian's claim** | E1 | 6 | Dep on previous mark |
### 2(b)
| Type II error. Accepted $H_0$ when $H_0$ was false (oe) | B1ft | Follow through on conclusion in (a) |
| --- | --- | --- |
| | E1 | 2 | Dep on previous mark |
| | | If Reject $H_0$ in (a) then: No error (B1fi) Rejected $H_0$ when $H_0$ was false (oe) (E1) |
| | | **8** | |2 The times taken to complete a round of golf at Slowpace Golf Club may be modelled by a random variable with mean $\mu$ hours and standard deviation 1.1 hours.
Julian claims that, on average, the time taken to complete a round of golf at Slowpace Golf Club is greater than 4 hours.
The times of 40 randomly selected completed rounds of golf at Slowpace Golf Club result in a mean of 4.2 hours.
\begin{enumerate}[label=(\alph*)]
\item Investigate Julian's claim at the $5 \%$ level of significance.
\item If the actual mean time taken to complete a round of golf at Slowpace Golf Club is 4.5 hours, determine whether a Type I error, a Type II error or neither was made in the test conducted in part (a). Give a reason for your answer.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2012 Q2 [8]}}