In Exercises 25–28, determine whether a normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions p1 and p2 at the level of significance α. Assume the samples are random and independent.


Claim: p1<p2; α=0.05


Sample statistics: x1 = 86, n1=900 and x2 = 107, n2 = 1200

Verified step by step guidance

1. Step 1: Verify the conditions for using a normal sampling distribution. To use a normal approximation for the difference between two population proportions, ensure that both sample sizes are large enough. Specifically, check that n1 * p̂1 ≥ 5, n1 * (1 - p̂1) ≥ 5, n2 * p̂2 ≥ 5, and n2 * (1 - p̂2) ≥ 5, where p̂1 = x1 / n1 and p̂2 = x2 / n2.
2. Step 2: Calculate the sample proportions p̂1 and p̂2 using the formulas p̂1 = x1 / n1 and p̂2 = x2 / n2. Substitute the given values: x1 = 86, n1 = 900, x2 = 107, and n2 = 1200.
3. Step 3: Compute the pooled proportion p̂ using the formula: p̂ = (x1 + x2) / (n1 + n2). This pooled proportion will be used to calculate the standard error of the difference between the two proportions.
4. Step 4: Calculate the standard error (SE) of the difference between the two proportions using the formula: SE = sqrt(p̂ * (1 - p̂) * (1/n1 + 1/n2)). Substitute the values for p̂, n1, and n2 into the formula.
5. Step 5: Perform the hypothesis test. Compute the test statistic z using the formula: z = (p̂1 - p̂2) / SE. Compare the calculated z-value to the critical z-value for α = 0.05 (one-tailed test). If z is less than the critical value, reject the null hypothesis and conclude that p1 < p2; otherwise, fail to reject the null hypothesis.

Key Concepts

Normal Sampling Distribution

A normal sampling distribution is a probability distribution of sample means or proportions that approximates a normal distribution as the sample size increases, according to the Central Limit Theorem. For proportions, this approximation is valid when both np and n(1-p) are greater than 5, ensuring that the sample size is sufficiently large to yield reliable results.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this context, it involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1, based on a specified significance level (α).

Difference Between Two Proportions

The difference between two proportions is a statistical measure used to compare the proportions of two different groups. In this case, it involves calculating the difference between the sample proportions p1 and p2, and assessing whether this difference is statistically significant using a z-test for proportions, which helps determine if the observed difference is likely due to chance.