The n=12 observations X have mean 9 and population variance 10. Define \(Y=3X+14\). A second disjoint group has 14 Y-values with mean 45.
Find the mean of Y. Explain your reasoning. [2]
Find the variance of Y. Explain your reasoning. [2]
Find the sum of Y in the first group. Explain your reasoning. [2]
Solution
Hint
The two groups are disjoint when combining observations. Apply the affine transformation to the mean.
Worked solution
(a)
\(E(Y)=3(9)+14=41\)
Answer: \(41\).
Answer: \(41\)
(b)
\(\operatorname{Var}(Y)=3^2(10)=90\)
Answer: \(90\).
Answer: \(90\)
(c)
\(\sum Y=12(41)=492\)
Answer: \(492\).
Answer: \(492\)
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 the affine transformation to the mean.
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: Square the multiplier when transforming variance.
(b) · 2 marks
1 mark: Square the scale factor for variance.
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: Weight group means by the number of observations.
(c) · 2 marks
1 mark: Multiply the first-group mean by its 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: Square the multiplier when transforming variance.