Hint
Categories cover the whole outcome space and do not overlap. Multiply the total by each null probability.
Worked solution
(a)
\(\displaystyle E= \frac{104}{4}= 26\)
Answer: \(26\).
Answer: \(26\)
(b)
\(\displaystyle \sum \frac{(O- E)^{2}}{E}= \frac{45}{13}\)
Answer: \(\displaystyle \frac{45}{13}\).
Answer: \(\displaystyle \frac{45}{13}\)
(c)
\(\displaystyle P(\chi^2_3\ge \frac{45}{13})=0.325784820275207\)
Answer: \(0.325784820275207\).
Answer: \(\approx 0.325784820275207\)
(d)
The p-value is \(0.325784820275207\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the specified categorical distribution does not fit. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.325784820275207\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the specified categorical distribution does not fit. 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: Multiply the total by each null probability.
- 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 expected, not observed, frequency in the denominator.
(b) · 2 marks
- 1 mark: Add all four squared standardised deviations.
- 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: Subtract the number of fitted parameters when determining degrees of freedom.
(c) · 2 marks
- 1 mark: Use an upper tail with four minus one 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 expected, not observed, frequency in the denominator.
(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.