Paper 2 · normal distribution
The volume of dish washing liquid in satchets is normally distributed with mean 400 ml and standard deviation 45 ml, independent of the volume of fabric softener in bottles, which is normally distributed with mean 650 ml and standard deviation 50 ml. Let W be the total volume of 2 randomly chosen bottles of fabric softener minus the total volume of 4 randomly chosen satchets of dish washing liquid. What is the standard deviation of W, in ml (to 1 decimal place)?
A67.3 ml
B114.5 ml
C118.5 ml
D280.0 ml
Explanation: Var(W) = 2(50^2) + 4(45^2) = 5000 + 8100 = 13100, since the variance of a sum or difference of independent normal variables is the sum of their individual variances, each scaled by the number of terms. SD(W) = sqrt(13100) is approximately 114.5 ml. Using only one bottle and one satchet gives 67.3 ml, swapping the two counts gives 118.5 ml, and adding scaled standard deviations directly instead of variances gives 280.0 ml.
Derived from ZIMSEC Statistics Paper 2, November 2022, Q1
