An aptitude test for deep-sea divers produces scores which are normally distributed on a scale from

Question

An aptitude test for deep-sea divers produces scores which are normally distributed on a scale from $0$ to $100$. A random sample of $160$ divers were assessed, and each of their individual test scores, $x$, was recorded. The results are summarised by

$\sum x = 9120, \qquad\qquad\qquad \sum (x-\bar{x})^{2} = 35775$.

Calculate unbiased estimates of the population mean and variance.
Assuming that the unbiased estimate of the population mean obtained above is equal to the population mean, find the value of $c$ such that there is a probability of $0.05$ that a sample of $160$ scores has a sample mean that differs from its mean score by more than $c$.
Comment on the validity of the calculations if the scores had not been known to be normally distributed.

Accepted Answer

(i) $\hat\mu=57,\ \hat\sigma^2=225$ (ii) $c=2.32\text{ (3 s.f.)}$ (iii) Approximately valid by the Central Limit Theorem because $n=160$ is large.

Assumptions For Sampling

Assumptions For Sampling

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