Hint
The two cumulative probabilities correspond to different z values. Subtract the two linear quantile equations.
Worked solution
(a)
\(\displaystyle \sigma = \frac{b- a}{z_{2}- z_{1}}= 3\)
Answer: \(3\).
Answer: \(3\)
(b)
\(\mu=a-z_1\sigma=35\)
Answer: \(35\).
Answer: \(35\)
(c)
\(\displaystyle 1- \Phi (\frac{40- 35}{3})= 0.0477903522728147\)
Answer: \(0.0477903522728147\).
Answer: \(\approx 0.0477903522728147\)
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: Subtract the two linear quantile equations.
- 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 second normal parameter may be variance rather than standard deviation.
(b) · 2 marks
- 1 mark: Substitute the recovered scale into one quantile equation.
- 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 average the two observed cutoffs unless their z values are symmetric.
(c) · 2 marks
- 1 mark: Use the fitted normal parameters in the upper tail.
- 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 second normal parameter may be variance rather than standard deviation.
Total: 6 marks.