Statistical Inference: A Short Course

10.10 Hypothesis Tests for The Population Proportion (of Successes) p

Suppose we extract a random sample of size n (>30) from a binomial population and we seek to test a hypothesis about p, the population proportion of successes. Under H₀: p = p_o, we know that, for a binomial random variable X,

Let us construct our standardized test statistic as

(10.1)

which is N(0,1) distributed for large n. Here Equation (10.1) serves as the calculated value of our test statistic under H₀: p = p_o (assumed true).¹

Looking to the types of tests, we have the following:

a. One-tail test (upper tail)

For α small, (Fig. 10.12).

b. One-tail test (lower tail)

Now, (Fig. 10.13).

c. Two-tail test

Figure 10.12 Upper-tail critical region.

Figure 10.13 Lower-tail critical region.

In this instance,