Hint
The normal model is stated; σ>0. Standardise the cutoff using the stated standard deviation.
Worked solution
(a)
\(\displaystyle \Phi (\frac{61.8- 57}{3})= \Phi (\frac{8}{5})= 0.945200708300442\)
Answer: \(0.945200708300442\).
Answer: \(\approx 0.945200708300442\)
(b)
\(\displaystyle \Phi(\frac{8}{5})-\Phi(-1)=0.786545454368985\)
Answer: \(0.786545454368985\).
Answer: \(\approx 0.786545454368985\)
(c)
\(\displaystyle x=57+3(\frac{8}{5})=61.8\)
Answer: \(61.8\).
Answer: \(\approx 61.8\)
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: Standardise the cutoff using the stated standard deviation.
- 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: Divide by σ, not \(\displaystyle \sigma ^{2}\), when standardising.
(b) · 2 marks
- 1 mark: Subtract cumulative probabilities at the two endpoints.
- 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: A right-tail probability needs 1−Φ(z).
(c) · 2 marks
- 1 mark: Convert the specified z-score back to the X scale.
- 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: Divide by σ, not \(\displaystyle \sigma ^{2}\), when standardising.
Total: 6 marks.