We now come to the heart of the subject of statistical inference. Up until now the following type of question has been examined: given the population parameters ¡1 and a2, what is the probability of the sample mean X, from a sample of size n, being greater than some specified value or within some range of values? The parameters ¡1 and a2 are assumed to be known and the objective is to try to form some conclusions about possible values of X. However, in practice it is usually the sample values X and s2 that are known, while the population parameters ¡1 and a2 are not. Thus a more interesting question to ask is: given the values of X and s2, what can be said about ¡1 and a2? Sometimes the population variance is known, and inferences have to be made about ¡1 alone. For example, if a sample of 50 British families finds an average weekly expenditure on food (X) of £37.50 with a standard deviation (s) of £6.00, what can be said about the average expenditure (¡) of all British families?

Schematically this type of problem is shown as follows:

Sample information |

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