Mathematics

Mathematics

Problem

A factory records the diameters (in mm) of 5050 bolts. The arithmetic mean of the diameters is 12.4 mm12.4\ \text{mm} and the arithmetic mean of the squares of the diameters is 155.96 mm2155.96\ \text{mm}^{2}. a. (i) Calculate the variance of the diameters. b. (ii) State whether the batch meets the production standard if the allowable variance must not exceed 2 mm22\ \text{mm}^{2}.
Intuitive explanation

The key here is connecting the variance formula to the two means you're given. Variance measures spread, and it equals the mean of the squares minus the square of the mean—that is, Var=E[X2]−(E[X])2\text{Var} = E[X^2] - (E[X])^2. Once you find the variance, the standard deviation is just its square root, which tells you how tightly the bolt diameters cluster around the average.

Intuitive explanation

The key here is connecting the variance formula to the two means you're given. Variance measures spread, and it equals the mean of the squares minus the square of the mean—that is, Var=E[X2]−(E[X])2\text{Var} = E[X^2] - (E[X])^2. Once you find the variance, the standard deviation is just its square root, which tells you how tightly the bolt diameters cluster around the average.

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