| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Unknown variance (t-distribution) |
| Difficulty | Standard +0.3 This is a straightforward confidence interval construction with known variance assumption (S2 standard), requiring calculation of sample mean, application of z-critical value, and a simple interpretation. The only mild challenge is recognizing the assumption needed and making a basic comment, but the mechanics are routine for this module. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Assumption: speeds of cars normally distributed | B1 | |
| \(\bar{x} = 35.6\) | B1 | |
| \(s^2 = 38.27\) (\(s = 6.186\)) | B1 | \(\sigma^2 = 34.44\) (\(\sigma = 5.869\)) |
| 99% CI: \(35.6 \pm 3.250 \times \frac{6.186}{\sqrt{10}}\) | B1 | or use of \(\frac{\sqrt{34.44}}{3}\) |
| \(= 35.6 \pm 6.36\) | M1, \(A1\checkmark\) | on their mean and standard deviation |
| \(= (29.2, 42.0)\) | A1 | CAO (29.24, 41.96) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Confidence interval includes 30 mph | \(B1\checkmark\) | |
| 80% of sample exceed 30 mph limit | B1 | |
| Speed limit not adhered to | B1 | dependent on previous B1 |
# Question 5(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Assumption: speeds of cars normally distributed | B1 | |
| $\bar{x} = 35.6$ | B1 | |
| $s^2 = 38.27$ ($s = 6.186$) | B1 | $\sigma^2 = 34.44$ ($\sigma = 5.869$) |
| 99% CI: $35.6 \pm 3.250 \times \frac{6.186}{\sqrt{10}}$ | B1 | or use of $\frac{\sqrt{34.44}}{3}$ |
| $= 35.6 \pm 6.36$ | M1, $A1\checkmark$ | on their mean and standard deviation |
| $= (29.2, 42.0)$ | A1 | CAO (29.24, 41.96) |
# Question 5(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Confidence interval includes 30 mph | $B1\checkmark$ | |
| 80% of sample exceed 30 mph limit | B1 | |
| Speed limit **not** adhered to | B1 | dependent on previous B1 |
---5 Members of a residents' association are concerned about the speeds of cars travelling through their village. They decide to record the speed, in mph , of each of a random sample of 10 cars travelling through their village, with the following results:
$$\begin{array} { l l l l l l l l l l }
33 & 27 & 34 & 30 & 48 & 35 & 34 & 33 & 43 & 39
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Construct a $99 \%$ confidence interval for $\mu$, the mean speed of cars travelling through the village, stating any assumption that you make.
\item Comment on the claim that a 30 mph speed limit is being adhered to by most motorists.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2007 Q5 [10]}}