Hint
Population variances are known and the two samples are independent. Add the two independent sampling variances.
Worked solution
(a)
\(\displaystyle \frac{\sigma_{X}^{2}}{n_{X}}+ \frac{\sigma_{Y}^{2}}{n_{Y}}= \frac{203}{144}\)
Answer: \(\displaystyle \frac{203}{144}\).
Answer: \(\displaystyle \frac{203}{144}\)
(b)
\(\displaystyle z= \frac{\bar{x}- \bar{y}}{S}E= 2.10558721903079\)
Answer: \(2.10558721903079\).
Answer: \(\approx 2.10558721903079\)
(c)
\(1-\Phi(z)=0.0176201119121373\)
Answer: \(0.0176201119121373\).
Answer: \(\approx 0.0176201119121373\)
(d)
The p-value is \(0.0176201119121373\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the first population mean is greater than the second. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.0176201119121373\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the first population mean is greater than the second. This does not give the probability that \(\displaystyle H_{0}\) is true.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Add the two independent sampling variances.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Divide each population variance by its own sample size.
(b) · 2 marks
- 1 mark: Standardise the difference under null mean zero.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not pool known population variances as if they were sample estimates.
(c) · 2 marks
- 1 mark: Use the upper standard-normal tail.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Divide each population variance by its own sample size.
(d) · 1 marks
- 1 mark: Compare the relevant p-value with 0.05 and state the contextual conclusion without claiming the null is proved.
Alternative methods: A comparison with the correct critical value and the same valid contextual conclusion is acceptable.
Common error: A p-value is not the probability that the null hypothesis is true.
Total: 7 marks.