Hint
Independence is needed to omit covariance terms in the variance. Apply linearity of expectation.
Worked solution
(a)
\(4(13)-13=39\)
Answer: \(39\).
Answer: \(39\)
(b)
\(4^2(3)+5=53\)
Answer: \(53\).
Answer: \(53\)
(c)
\(\displaystyle 1- \Phi (\frac{42- 39}{\sqrt{53}})= 0.340139773667225\)
Answer: \(0.340139773667225\).
Answer: \(\approx 0.340139773667225\)
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.