Hint
The samples are independent and observations have no ties in these exact examples. Add the pooled ranks assigned to A.
Worked solution
(a)
\(R=14\)
Answer: \(14\).
Answer: \(14\)
(b)
\(\displaystyle U= R- \frac{n_{1}(n_{1}+ 1)}{2}= 4\)
Answer: \(4\).
Answer: \(4\)
(c)
\(\displaystyle P(R\le 14)=\frac{2}{55}\) by equally likely choices of 4 pooled ranks.
Answer: \(\displaystyle \frac{2}{55}\).
Answer: \(\displaystyle \frac{2}{55}\)
(d)
The p-value is \(0.0363636363636364\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the first population tends to give smaller values. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.0363636363636364\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the first population tends to give smaller values. 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 pooled ranks assigned to A.
- 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 rank each sample separately.
(b) · 2 marks
- 1 mark: Subtract the smallest possible rank sum for a sample of this size.
- 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 apply a paired procedure to independent samples.
(c) · 2 marks
- 1 mark: Enumerate selections of A ranks under the identical-distribution null.
- 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 rank each sample separately.
(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.