Statistical Inference: A Short Course

7.4 A Central Limit Theorem

Up to this point in our discussion of the sampling distribution of the mean, we have not made any assumption about the form (shape) of the parent population. For convenience, let us assume that we are sampling from an infinite population. In this regard:

1. Suppose that our population variable X is N(μ, σ). Then will be and thus the standardized sample mean

Hence we can use the standard normal area table to calculate probabilities involving For instance, given that X is N(20,15) and n = 100, find To this end, we have

(See Fig. 7.4).

2. Suppose X is “not” N(μ, σ). Can we still use the N(0, 1) area table to calculate probabilities involving ? Interestingly enough, the answer is “yes” if n is large enough. Large enough” means n > 30. The “yes” answer is provided ...