Hint
A normal population is assumed for exact small-sample t inference. Use the estimated standard error of the sample mean.
Worked solution
(a)
\(\displaystyle t= \frac{\bar{x}- \mu_{0}}{\frac{s}{\sqrt{n}}}= 1.05830052442584\)
Answer: \(1.05830052442584\).
Answer: \(\approx 1.05830052442584\)
(b)
\(n-1=6\)
Answer: \(6\).
Answer: \(6\)
(c)
\(P(T_{6}\ge 1.05830052)=0.165326456913808\)
Answer: \(0.165326456913808\).
Answer: \(\approx 0.165326456913808\)
(d)
The p-value is \(0.165326456913808\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the population mean exceeds its null value. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.165326456913808\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the population mean 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: Use the estimated standard error of the sample mean.
- 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: Use n−1 degrees of freedom, not n.
(b) · 2 marks
- 1 mark: Account for estimating one mean.
- 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: Keep the one-sided test tail distinct from a two-sided interval.
(c) · 2 marks
- 1 mark: Use the upper t tail with n−1 degrees of freedom.
- 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: Use n−1 degrees of freedom, not n.
(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.