Hint
The pairing is meaningful and fixed before analysis. Keep the specified subtraction order.
Worked solution
(a)
\(\displaystyle \bar{d}= \sum \frac{d}{6}= 17\)
Answer: \(17\).
Answer: \(17\)
(b)
\(\displaystyle s_{d}^{2}= \sum \frac{(d- \bar{d})^{2}}{5}= \frac{378}{5}\)
Answer: \(\displaystyle \frac{378}{5}\).
Answer: \(\displaystyle \frac{378}{5}\)
(c)
\(\displaystyle t= \frac{\bar{d}}{\frac{s_{d}}{\sqrt{6}}}= 4.78920744346159\)
Answer: \(4.78920744346159\).
Answer: \(\approx 4.78920744346159\)
(d)
The p-value is \(0.00246518477442957\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the mean paired difference is positive. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.00246518477442957\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the mean paired 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: Keep the specified subtraction order.
- 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 use the two-sample standard error for paired observations.
(b) · 2 marks
- 1 mark: Use the five residual 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: Reversing the subtraction order reverses both the sign and the relevant alternative.
(c) · 2 marks
- 1 mark: Treat the six differences as one sample.
- 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 use the two-sample standard error for paired observations.
(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.