Give a reason why sampling would be required in order to reach a conclusion about
the mean height of adult males in England,
the mean weight that can be supported by a single cable of a certain type without the cable breaking.
The weights, in kg , of sacks of potatoes are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The weights of a random sample of 500 sacks of potatoes are found and the results are summarised below.
$$n = 500 , \quad \Sigma x = 9850 , \quad \Sigma x ^ { 2 } = 194125 .$$
Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
A further random sample of 60 sacks of potatoes is taken. Using your values from part (b) (i), find the probability that the mean weight of this sample exceeds 19.73 kg .
Explain whether it was necessary to use the Central Limit Theorem in your calculation in part (b) (ii).