Hint
Check that expected success and failure counts are not small. Divide successes by the sample size.
Worked solution
(a)
\(\displaystyle \hat{p}= \frac{53}{180}= \frac{53}{180}\)
Answer: \(\displaystyle \frac{53}{180}\).
Answer: \(\displaystyle \frac{53}{180}\)
(b)
\(\displaystyle \sqrt{\frac{\hat{p}(1- \hat{p})}{n}}= 0.0339727761581279\)
Answer: \(0.0339727761581279\).
Answer: \(\approx 0.0339727761581279\)
(c)
\(\hat p-z\,SE=0.227859026719673\)
Answer: \(0.227859026719673\).
Answer: \(\approx 0.227859026719673\)
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: Divide successes by the sample size.
- 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 null-hypothesis \(\displaystyle p_{0}\) as the estimate when constructing this interval.
(b) · 2 marks
- 1 mark: Use the estimated Bernoulli variance divided by n.
- 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: The interval width is twice the margin of error.
(c) · 2 marks
- 1 mark: Subtract the two-sided normal margin.
- 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 null-hypothesis \(\displaystyle p_{0}\) as the estimate when constructing this interval.
Total: 6 marks.