Hint
Independent observations each belong to exactly one cell. Multiply the row and column totals and divide by the grand total.
Worked solution
(a)
\(\displaystyle \frac{R_{1}C_{1}}{n}= 27\)
Answer: \(27\).
Answer: \(27\)
(b)
\(\displaystyle \sum_{ij} \frac{(O_{ij}- E_{ij})^{2}}{E_{ij}}= 3.7037037037037\)
Answer: \(3.7037037037037\).
Answer: \(\approx 3.7037037037037\)
(c)
\(P(\chi^2_2\ge 3.7037037037037)=0.156946255820714\)
Answer: \(0.156946255820714\).
Answer: \(\approx 0.156946255820714\)
(d)
The p-value is \(0.156946255820714\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the two classifications are associated. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.156946255820714\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the two classifications are associated. 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 row and column totals and divide by the grand total.
- 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 row and column totals, not an average of observed cell counts.
(b) · 2 marks
- 1 mark: Sum all six cell contributions.
- 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: An association is not a causal conclusion.
(c) · 2 marks
- 1 mark: Use (2−1)(3−1) 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 row and column totals, not an average of observed cell counts.
(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.