Hint
Pairs or observations are independently sampled. Use success probability one half under the null median.
Worked solution
(a)
\(\displaystyle \frac{n}{2}= 4\)
Answer: \(4\).
Answer: \(4\)
(b)
\(\displaystyle P(\operatorname{Bin}(8, \frac{1}{2})\ge 6)= 0.14453125\)
Answer: \(0.14453125\).
Answer: \(\approx 0.14453125\)
(c)
\(\min(1,2p_{\rm one-sided})=0.2890625\)
Answer: \(0.2890625\).
Answer: \(\approx 0.2890625\)
(d)
The p-value is \(0.14453125\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the population median difference is positive. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.14453125\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the population median difference is positive. 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: Use success probability one half under the null median.
- 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 magnitudes for a sign test.
(b) · 2 marks
- 1 mark: Include the observed positive count and all larger counts.
- 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: A two-sided p-value must account for both extreme directions.
(c) · 2 marks
- 1 mark: Double the symmetric smaller binomial 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: Do not rank magnitudes for a sign test.
(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.