Hint
The alternative direction and critical region must match. Calculate the upper binomial tail under \(\displaystyle p_{0}\).
Worked solution
(a)
\(P_{0.3}(X\ge 9)=0.015242525769771\)
Answer: \(0.015242525769771\).
Answer: \(\approx 0.015242525769771\)
(b)
\(P_{0.5}(X\ge 9)=0.303619384765625\)
Answer: \(0.303619384765625\).
Answer: \(\approx 0.303619384765625\)
(c)
\(1-\operatorname{power}=0.696380615234375\)
Answer: \(0.696380615234375\).
Answer: \(\approx 0.696380615234375\)
(d)
The p-value is \(0.003652521008436\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the success probability exceeds its null value. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.003652521008436\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the success probability exceeds its null value. 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: Calculate the upper binomial tail under \(\displaystyle p_{0}\).
- 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 calculate the null p-value using the observed proportion.
(b) · 2 marks
- 1 mark: Evaluate the same critical region under \(\displaystyle p_{1}\).
- 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 requested 5% discrete test may have actual size below 5%.
(c) · 2 marks
- 1 mark: Take the probability of the acceptance region under \(\displaystyle p_{1}\).
- 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 calculate the null p-value using the observed proportion.
(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.