Question

Please explain how the proportions for two populations are used in hypotheses testing about two population proportions. Please give an example.

Answer 1

We use z test for checking the significant difference between the two population proportions. For this hypothesis test, we use the sample proportions as an estimate for the population proportions. Let us see one simple example:

Suppose, we want to check the claim whether the proportion of graduate men and women is same or not. Suppose we take a random sample of 20 men and it is found that 5 of them are graduate. Also, we take a random sample of 15 women and it is found that 4 of them are graduate.

The null and alternative hypotheses are given as below:

Null hypothesis: H₀: The proportion of graduate men and women is same.

Alternative hypothesis: H_a: The proportion of graduate men and women is not same.

That is, we have

H₀: p₁ = p₂

Versus

H_a: p₁ ≠ p₂

This is a two tailed test.

Test statistic for z test for two population proportions is given as below:

Z = (P1 – P2) / sqrt(P*(1 – P)*((1/N1) + (1/N2)))

Estimates for population proportions are given as below:

X1 = 5, N1 = 20, P1 = X1/N1 = 5/20 = ¼ = 0.25

X2 = 4, N2 = 15, P2=X2/N2 = 4/15 = 0.266667

P = (X1+X2)/(N1+N2) = (5+4)/(20+15) = 0.2571

Z = (P1 – P2) / sqrt(P*(1 – P)*((1/N1) + (1/N2)))

Z = (0.25 – 0.266667) / sqrt(0.2571*(1 – 0.2571)*((1/20) + (1/15)))

Z = -0.1116

P-value = 0.9111

(by using z-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that the proportion of graduate men and women is same.