| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| 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 one-sample z-test with clearly stated hypotheses (testing if mean has decreased from 44.1 mph). The calculations are routine: find sample mean, use given variance, apply standard normal test. Part (c) on error types is standard bookwork. Slightly above average only because it requires interpreting Type I/II errors in context, but this is expected S2 material with no novel problem-solving required. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| The speeds of vehicles are normally distributed | B1 | Accept equivalent statements about the population/underlying distribution being normal |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu = 44.1\) | B1 | Both hypotheses required |
| \(H_1: \mu < 44.1\) | One-tailed test | |
| \(\bar{x} = \frac{4327.0}{100} = 43.27\) | B1 | |
| \(s^2 = \frac{925.71}{99} = 9.3506...\) or \(s = 3.0579...\) | M1 | Attempt at unbiased estimate of variance using \(\frac{\sum(x-\bar{x})^2}{n-1}\) |
| \(t = \frac{43.27 - 44.1}{\frac{s}{\sqrt{100}}} = \frac{-0.83}{0.30579...}\) | M1 | Correct structure for test statistic |
| \(t = -2.715...\) | A1 | Accept \(-2.71\) to \(-2.72\) |
| Critical value: \(t_{99}\) at 1% one-tailed \(= -2.365\) (or \(2.365\)) | B1 | Accept \(t_{99}\) or \(t_{100}\) giving \(2.364\) or \(2.326\) |
| \( | -2.715 | > 2.365\), reject \(H_0\) |
| There is sufficient evidence at the 1% level that the mean speed has been reduced | A1 | In context, follow through on their test statistic |
| Answer | Marks | Guidance |
|---|---|---|
| A Type I error is concluding that the mean speed has been reduced (i.e. rejecting \(H_0\)) when in fact it has not been reduced (i.e. \(H_0\) is true, mean is still 44.1 mph) | B1 | Must be in context |
| Answer | Marks | Guidance |
|---|---|---|
| A Type II error is concluding that the mean speed has not been reduced (i.e. not rejecting \(H_0\)) when in fact it has been reduced (i.e. \(H_1\) is true, mean is less than 44.1 mph) | B1 | Must be in context |
# Question 4:
## Part (a)
| The speeds of vehicles are normally distributed | B1 | Accept equivalent statements about the population/underlying distribution being normal |
## Part (b)
| $H_0: \mu = 44.1$ | B1 | Both hypotheses required |
| $H_1: \mu < 44.1$ | | One-tailed test |
| $\bar{x} = \frac{4327.0}{100} = 43.27$ | B1 | |
| $s^2 = \frac{925.71}{99} = 9.3506...$ or $s = 3.0579...$ | M1 | Attempt at unbiased estimate of variance using $\frac{\sum(x-\bar{x})^2}{n-1}$ |
| $t = \frac{43.27 - 44.1}{\frac{s}{\sqrt{100}}} = \frac{-0.83}{0.30579...}$ | M1 | Correct structure for test statistic |
| $t = -2.715...$ | A1 | Accept $-2.71$ to $-2.72$ |
| Critical value: $t_{99}$ at 1% one-tailed $= -2.365$ (or $2.365$) | B1 | Accept $t_{99}$ or $t_{100}$ giving $2.364$ or $2.326$ |
| $|-2.715| > 2.365$, reject $H_0$ | M1 | Correct comparison and conclusion |
| There is sufficient evidence at the 1% level that the mean speed has been reduced | A1 | In context, follow through on their test statistic |
## Part (c)(i)
| A Type I error is concluding that the mean speed has been reduced (i.e. rejecting $H_0$) when in fact it has not been reduced (i.e. $H_0$ is true, mean is still 44.1 mph) | B1 | Must be in context |
## Part (c)(ii)
| A Type II error is concluding that the mean speed has not been reduced (i.e. not rejecting $H_0$) when in fact it has been reduced (i.e. $H_1$ is true, mean is less than 44.1 mph) | B1 | Must be in context |4 Wellgrove village has a main road running through it that has a 40 mph speed limit. The villagers were concerned that many vehicles travelled too fast through the village, and so they set up a device for measuring the speed of vehicles on this main road. This device indicated that the mean speed of vehicles travelling through Wellgrove was 44.1 mph .
In an attempt to reduce the mean speed of vehicles travelling through Wellgrove, life-size photographs of a police officer were erected next to the road on the approaches to the village. The speed, $X \mathrm { mph }$, of a sample of 100 vehicles was then measured and the following data obtained.
$$\sum x = 4327.0 \quad \sum ( x - \bar { x } ) ^ { 2 } = 925.71$$
\begin{enumerate}[label=(\alph*)]
\item State an assumption that must be made about the sample in order to carry out a hypothesis test to investigate whether the desired reduction in mean speed had occurred.
\item Given that the assumption that you stated in part (a) is valid, carry out such a test, using the $1 \%$ level of significance.
\item Explain, in the context of this question, the meaning of:
\begin{enumerate}[label=(\roman*)]
\item a Type I error;
\item a Type II error.\\[0pt]
[2 marks]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2015 Q4 [11]}}