Hint
Independence is needed to omit covariance terms in the variance. Apply linearity of expectation.
Worked solution
(a)
\(2(14)-15=13\)
Answer: \(13\).
Answer: \(13\)
(b)
\(2^2(4)+3=19\)
Answer: \(19\).
Answer: \(19\)
(c)
\(\displaystyle 1- \Phi (\frac{14- 13}{\sqrt{19}})= 0.409272904191022\)
Answer: \(0.409272904191022\).
Answer: \(\approx 0.409272904191022\)
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: Apply linearity of expectation.
- 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 negative coefficient does not subtract an independent variance.
(b) · 2 marks
- 1 mark: Square both coefficients and add independent variances.
- 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 confuse one value multiplied by a constant with several independent values.
(c) · 2 marks
- 1 mark: Use the normal distribution of the independent linear combination.
- 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 negative coefficient does not subtract an independent variance.
Total: 6 marks.