Probability

THE SAMPLING DISTRIBUTION OF THE MEAN (σ KNOWN)

Suppose a random sample of n observations has been taken from some population and that x has been computed, say, to estimate the mean of thepopulation. It should be clear that if we took a second random sample of size n from this population, it would be quite unreasonable to expect the identical value for x, and if we took several more samples,probably no two of the x’s would be alike.

To approach this question experimentally, let us actually perform a simple experiment in which 50 random samples of size n = 10 are taken from apopulation having the discrete uniform distribution.
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The means we actually obtained in these 50 samples are:
4.4 3.2 5.0 3.5 4.1 4.4 3.6 6.5 5.3 4.4
3.1 5.33.8 4.3 3.3 5.0 4.9 4.8 3.1 5.3
3.0 3.0 4.6 5.8 4.6 4.0 3.7 5.2 3.7 3.8
5.3 5.5 4.8 6.4 4.9 6.5 3.5 4.54.9 5.3
3.6 2.7 4.0 5.0 2.6 4.2 4.4 5.6 4.7 4.3

And the following is a frequency table showing the distribution of these means:

2.0-2.92
3.0-3.9 14
4.0-4.9 19
5.0-5.9 12
6.0-6.9 3
50

Would we get a similardistribution if we took 100 samples, 1000 samples, or perhaps even more? To answer this kind of question, we shall have to investigate the theoretical sampling distribution of x which, for the given example,provides us with the probabilities of getting x’s between 2.0 and 2.9, between 3.0 and 3.9… between 6.0 and 6.9, and perhaps values less than 2.0 or greater than 6.9.

THEOREM 7.1. If a random sampleof size n is taken from a population having the mean µ and the variance σ2, then x is a value of a random variable whose distribution has the mean µ. For samples from infinite populations the...