A2 June 2024 Q3
3. A factory produces bolts. The lengths of the bolts are normally distributed with mean \(\mu\) mm and standard deviation 0.868 mm
A random sample of 15 of these bolts is taken and the mean length is 30.03 mm
A suitable test, at the 10% level of significance, is carried out using these 15 bolts, to see whether or not there is evidence that the variance of the length of the bolts has increased.
The manager of the factory decides that, in future, he will check each month whether the machine making the bolts is working properly. He uses a 10% level of significance to test whether or not there is evidence that
- the mean length of the bolts has changed
- the variance of the length of the bolts has increased
The next month a random sample of 15 bolts is taken.
The mean length of these bolts is 30.06 mm and the standard deviation is 1.02 mm
Give reasons for your answer. (2)
| Scheme | Marks | AO |
|---|---|---|
| 1.6449 | B1 | 3.3 |
| \(30.03 \pm \dfrac{0.868}{\sqrt{15}} \times \text{“}1.6449\text{”}\) | M1 | 2.1 |
| \((29.6613\ldots,\ 30.3986\ldots)\) | A1 | 1.1b |
| (3) |
Notes
B1: For realising a normal distribution must be used as a model and finding the correct value 1.6449 or better.
M1: For \(30.03 \pm \dfrac{0.868}{\sqrt{15}} \times \text{“}z\ \text{value}\text{”}\). May be implied by a correct CI
A1: awrt 29.7 and 30.4 from correct working (B0M1A1 is possible)
| Scheme | Marks | AO |
|---|---|---|
| \(\chi^2_{14}(0.1)\) CV \(= 21.064\) | B1 | 3.3 |
| (Reject H0 if) \(\dfrac{14S^2}{0.868^2} \gt 21.06\) | M1 | 2.1 |
| Critical region is \(S^2 \gt 1.133\ldots\) | A1 | 1.1b |
| (3) |
Notes
B1: For realising a chi squared distribution must be used as a model and CV = awrt 21.06
M1: Correct method comparing \(\dfrac{14S^2}{0.868^2}\) to \(19 \lt \chi^2_{14} \lt 24\) (condone equals instead of >)
Ignore any additional calculations outside of this range.
A1: Correct CR allow awrt 1.13 only
| Scheme | Marks | AO |
|---|---|---|
| Insufficient evidence that the machine is not working properly as 30.06 is within the confidence interval | M1 | 2.2b |
| and 1.022 (1.0404) is not in the CR oe | A1ft | 2.4 |
| (2) | ||
| (8 marks) |
Notes
M1: Dependent on an answer to part (a) or part (b) in order to draw an inference. Drawing a correct inference that there is insufficient evidence to suggest that the machine is not working properly following through one of their CR or CI with one correct comparison. Must mention the machine at least once.
If they have either 30.06 outside their CI or 1.022 in their CR then allow an inference that the machine is not working properly for this mark following a correct comparison and no contradictory statements relating to that comparison.
This mark can still be scored if there is a second comparison which is incorrect.
A1ft: Dependent on 30.06 in their confidence interval and 1.022 not in the CR. For drawing a correct inference following through their CR or CI with both correct comparisons and no contradictory statements. Must mention the machine at least once. Do not accept a comparison of 30.06 with just one of the limits for the CI.