S3 June 2014 (R) Q7
7. A petrol pump is tested regularly to check that the reading on its gauge is accurate. The random variable \(X\), in litres, is the quantity of petrol actually dispensed when the gauge reads 10.00 litres. \(X\) is known to have distribution \(X \sim \mathrm{N}(\mu, 0.08^2)\)
(a) Eight random tests gave the following values of \(x\)\[10.01 \quad 9.97 \quad 9.93 \quad 9.99 \quad 9.90 \quad 9.95 \quad 10.13 \quad 9.94\]
(i) Find a 95% confidence interval for \(\mu\) to 2 decimal places.
(ii) Use your result to comment on the accuracy of the petrol gauge.
(5)(b) A sample mean of 9.96 litres was obtained from a random sample of \(n\) tests. A 90% confidence interval for \(\mu\) gave an upper limit of less than 10.00 litres. Find the minimum value of \(n\). (5)
| Scheme | Marks |
|---|---|
| (i) \(\bar{x} = \frac{10.01 + 9.97 + 9.93 + \cdots}{8} = 9.9775\) | M1 |
| 95% CI \(\bar{x} \pm 1.96 \times \frac{0.08}{\sqrt{8}}\) | B1M1 |
| 95% CI for \(\mu\) (9.92, 10.03) | A1 |
| (4) | |
| (ii) 10.00 is within confidence interval so accept that pump may be performing correctly (although sample mean is low). | B1 |
| (1) |
Notes
1st M1 attempt to find sample mean
B1 for correct z value
A1 limits correct to 2 decimal places (or more)
| Scheme | Marks |
|---|---|
| Upper limit of CI is \(9.96 + 1.6449 \times \dfrac{0.08}{\sqrt{n}} \lt 10.00\) \(\dfrac{1.6449 \times 0.08}{\sqrt{n}} \lt 0.04\) | B1, M1A1ft |
| \(\sqrt{n} \gt \dfrac{1.6449 \times 0.08}{0.04}\) | M1 |
| \(n \gt 10.82\ldots\) therefore minimum \(n = 11\) | A1 cao |
| (5) | |
| (10 marks) |
Notes
B1 for correct z value
1st M1A1, ft their z value