Random sampling in normal populations

In the last section, we drew several important conclusions without restricting the nature of the distribution of the observations in any way. We only had to assume that the samples were random. In this section we shall add another premise that the samples also come from normal populations. As we have already commented, this hypothesis is perfectly acceptable in many real situations, because of the central limit theorem. Under these two restrictions — random samples taken from normal populations — the sample values follow specific distributions that can be used to calculate confidence intervals. Before showing how this can be done, we shall state, without proof, some conclusions we shall need. If you are interested, you can find the proofs in a book on advanced statistics, such as Dudewicz and Mishra (1985). 

Let us consider samples of n elements, randomly drawn from a normal population of mean p and variance It can be shown that the sample values X and obey the following  

X = random variable distributed as N p, t ) (x, s ) = estimates of (ji, obtained from random samples of n elements tn-i = t distribution with n—1 degrees of freedom Xn-i = distribution with n—1 degrees of freedom. 

From these relations we can obtain confidence intervals, using arguments similar to those of Section 2.3. To illustrate, we shall again use a sample of 10 beans drawn from the 1-kg package. 

Suppose that the masses of these beans are the first 10 values in Table 2.2. For this sample, we already know (if you did Exercise 2.6) that X = 0.1887 gand s = 0.0423 g. If the masses follow a normal distribution, the sample average x is also normally distributed according to Eq. (2.16). Subtracting the population mean p and dividing the result by its standard deviation, aj3delds a standard normal variable  

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