Question

You obtain a sample from the first population with 202 successes and 103 failures. You obtain a sample from the second population with 557 successes and 188 failures. Suppose that all assumptions are met, so that the difference in sample proportions follows a normal distribution. What is the test statistic for this sample?

Answer 1

The test statistic for comparing two sample proportions is calculated using the pooled sample proportion and the standard error formula for the difference between proportions, then determining the Z-score.

Step-by-step explanation:

To calculate the test statistic for comparing two sample proportions, we need to use the formula for the standard error of the difference between two proportions, and then find the Z-score.

First, we calculate the sample proportions: For the first population, p1 = successes / (successes + failures) = 202 / (202 + 103). For the second population, p2 = 557 / (557 + 188).

Let's denote p1_hat and p2_hat as the observed sample proportions. The pooled sample proportion (p_pool) is calculated using both samples:

p_pool = (202 + 557) / ((202 + 103) + (557 + 188))

The standard error (SE) of the difference between the two sample proportions is:

SE = sqrt(p_pool * (1 - p_pool) * (1/(202 + 103) + 1/(557 + 188)))

Finally, the test statistic (Z) is calculated as:

Z = (p1_hat - p2_hat) / SE

Substitute the calculated values of p1_hat, p2_hat, and SE into the formula to find the test statistic, which will allow you to determine whether there is a significant difference between the two sample proportions.