Question

If random samples of the given sizes are drawn from populations with the given proportions: (a) Find the standard error of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\) (b) Determine whether the sample sizes are large enough for the Central Limit Theorem to apply. Samples of size 500 from population \(A\) with proportion 0.58 and samples of size 200 from population \(B\) with proportion 0.49

Short answer

The standard error of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\), is approximately 0.050. The sample sizes are large enough for the Central Limit Theorem to apply.

Step-by-step solution

Calculate Standard Error

First, plug the given population proportions (0.58 and 0.49) and sample sizes (500 and 200) into the formula for the standard error of the distribution of differences in sample proportions. As such, this would result in \(\sqrt{[(0.58)(1 - 0.58) / 500] + [(0.49)(1 - 0.49) / 200]}\). Calculate this expression to find the standard error.

Determine the Central Limit Theorem Applicability

To find if the Central Limit Theorem applies, check if both \(500 * 0.58 * (1 - 0.58) \geq 10\) and \(200 * 0.49 * (1 - 0.49) \geq 10\). If both of these inequalities hold true, then the sample sizes are large enough for the Central Limit Theorem to apply.