February 12, 2009 marked the 200th anniversary of Charles Darwin's birth. To celebrate, Gallup, a national polling organization, surveyed 1,018 randomly selected American adults about their education level and their beliefs about the theory of evolution. In their sample, 325 of their respondents had some college education and 228 were college graduates. Among the 325 respondents with some college education, 133 said that they believed in the theory of evolution. Among the 228 respondents who were college graduates, 121 said that they believed in the theory of evolution.

We want to test, at the 10% level, if there is evidence that the proportion of college graduates that believe in evolution differs significantly from the proportion of individuals with some college education that believe in evolution. Assume that the Pooled proportion (for standard error): = 0.459.
What is the z test statistic for this hypothesis test?

The objective of the poll is to compare the proportions of people who believe in the theory of evolution. For this, the investigator took a sample of 1018 and divided it into two groups using a criteria their level in college education.

Group 1 (people with some college education)

n₁= 325

x₁= 133

sample proportion ^ρ₁=x₁/n₁ = 133/325 = 0.409

Group 2 (college graduates)

n₂= 228

x₂= 121

sample proportion ^ρ₂=x₂/n₂ = 121/228 = 0.531

pooled proportion ^ρ= 0.459

The statistic test that corresponds to test if the proportions are equal or different is the difference of population proportions and you have to use the normal approximation for the proportions to calculate it.

The hypothesis is:

H₀: ρ₂=ρ₁

H₁: ρ₂≠ρ₁

α: 0.10

Z= (^ρ₂ - ^ρ₁) - (ρ₂ - ρ₁) ≈ N (0;1)

√(^ρ(1 - ^ρ)*(1/n₂+1/n₁)

The critical region is two-tailed, so you have a lower critrical value and an upper critical value:

[tex]Z_{\alpha /2} = Z_{0.05} = -1.64[/tex]

[tex]Z_{1-\alpha /2} = Z_{0.95} = 1.64[/tex]

If [tex]Z_{H0}[/tex] ≤ -1.64 or if [tex]Z_{H0}[/tex] ≥ 1.64 you reject the null hypothesis.

If -1.64 < [tex]Z_{H0}[/tex] < 1.64, the you do not reject the null hypothesis.

Z= (^ρ₂ - ^ρ₁) - (ρ₂ - ρ₁) = (0.531 - 0.409) - 0 = 2.83

√(^ρ(1 - ^ρ)*(1/n₂+1/n₁) √(0.459(1 - 0.459)*(1/228+1/325)

[tex]Z_{H0}[/tex] = 2.83

Since the calculated Z value is greater than the upper critical value, the decision is to reject the null hypothesis. In other words, there is significative enough evidence to say that there is a difference between the population proportion of people who believe in the evolution theory and graduated from college and the population proportion of people who believe in the evolution theory and have some college education.

I hope it helps!