June 2022 Paper 3 Q17
17 The number of working hours per week of employees in a company is modelled by a normal distribution with mean of 34 hours and a standard deviation of 4.5 hours.
The manager claims that the mean working hours per week of the company’s employees has increased.
A random sample of 30 employees in the company was found to have mean working hours per week of 36.2 hours.
Carry out a hypothesis test at the 2.5% significance level to investigate the manager’s claim. [6 marks]
| Scheme | Marks | AO |
|---|---|---|
| States both hypotheses correctly for one-tailed test | B1 | 2.5 |
| States or uses correct model PI by mean 34 and variance \(\dfrac{4.5^2}{30}\) or standard deviation \(\dfrac{4.5}{\sqrt{30}}\) OE or by correct probability AWFW [0.0037, 0.004] or test statistic \(\pm\dfrac{36.2 - 34}{4.5 \div \sqrt{30}}\) or test statistic value AWFW \(\pm\)[2.67, 2.68] or critical region > AWFW [35.6, 35.632] condone ≥ or critical value \(\pm\)[35.6, 35.632] | M1 | 1.1a |
| Obtains AWFW [0.0037, 0.004] or obtains the correct value of the test statistic AWFW [2.67, 2.68] or obtains the correct critical region > AWFW [35.6, 35.632] condone ≥ condone incorrect missing or incorrect labels | A1 | 1.1b |
| Compares their [0.0037, 0.004] with 0.025 or compares their [2.67, 2.68] with 1.96 allow positive values up to 4 for 1.96 or compares their region > [35.6, 35.632] with 36.2 condone ≥ | M1 | 3.5a |
| Infers \(H_0\) or null hypothesis rejected Condone \(H_1\) or alternative hypothesis accepted All figures must be correct | A1 | 2.2b |
| Concludes correctly in context that there is sufficient evidence to suggest that the mean working hours have increased To be awarded R1, marks M1A1M1A1 must be scored as the minimum | R1 | 3.2a |
| (6 marks) |
Typical solution
\(X\) = working hours per week
\[H_0: \mu = 34\]\[H_1: \mu \gt 34\]\[\bar{X} \sim \mathrm{N}\left(34, \frac{4.5^2}{30}\right)\]\[P(\bar{X} \gt 36.2) = 0.004\]\[0.004 \lt 0.025\]Reject \(H_0\)
There is sufficient evidence to suggest that the mean working hours have increased.