Question

Assume that you plan to use a significance level of α = 0.05 to test the claim that p1 = p2. Use the given sample sizes and numbers of successes to find the z test statistic for the hypothesis test.

n1 = 151 n2 = 144
x1 = 43x2 = 37

Answer 1

The z-test statistic is 1.2127. We fail to reject the null hypothesis. There is not enough evidence to conclude that the two population proportions are different at the 5% significance level.

Define the null and alternative hypotheses.

Null hypothesis (H₀): p₁ = p₂ (the two population proportions are equal).

Alternative hypothesis (H₁): p₁ ≠ p₂ (the two population proportions are not equal).

The pooled proportion.

`\bar p` = (x₁ + x₂) / (n₁ + n₂) = (78 + 69) / (190 + 184) = 0.7375

The standard error of the difference in proportions.

SE(p₁ - p₂) =
`√([\bar p(1 - \bar p) / n_1 + \bar p(1 - \bar p) / n_2])`

= √[0.7375(1 - 0.7375) / 190 + 0.7375(1 - 0.7375) / 184]

≈ 0.0426

The z-test statistic.

z = (p₁ - p₂) / SE(p₁ - p₂)

= (0.78 - 0.7375) / 0.0426

≈ 1.2127

Interpret the results.

The z-test statistic is 1.2127, which is not greater than the critical value of 1.96 for a two-tailed test at a significance level of 0.05.

Therefore, we fail to reject the null hypothesis.

There is not enough evidence to conclude that the two population proportions are different at the 5% significance level.