Hint
Categories cover the whole outcome space and do not overlap. Multiply the total by each null probability.
Worked solution
(a)
\(\displaystyle E= \frac{116}{4}= 29\)
Answer: \(29\).
Answer: \(29\)
(b)
\(\displaystyle \sum \frac{(O- E)^{2}}{E}= \frac{360}{29}\)
Answer: \(\displaystyle \frac{360}{29}\).
Answer: \(\displaystyle \frac{360}{29}\)
(c)
\(\displaystyle P(\chi^2_3\ge \frac{360}{29})=0.00609210677782619\)
Answer: \(0.00609210677782619\).
Answer: \(\approx 0.00609210677782619\)
(d)
The p-value is \(0.00609210677782619\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is 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 \(0.00609210677782619\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is 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.