A random sample of 1,556 people in country A were asked to...

A random sample of 1,556 people in country A were asked to respond to this statement: Increased world trade can increase our per capita prosperity. Of these sample members, $38.4 \%$ agreed with the statement. When the same statement was presented to a random sample of 1,108 people in country B, $52.0 \%$ agreed. Test the null hypothesis that the population proportions agreeing with this statement were the same in the two countries against the alternative that a higher proportion agreed in country $B$.

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis represents the status quo or a statement of no effect—in this scenario, that the population proportions are equal in both countries. The alternative hypothesis is what the test seeks to provide evidence for, stating that there is a difference, specifically that one country has a higher proportion than the other.

Two-Sample Proportion Test

This test is used to determine whether there is a statistically significant difference between the proportions from two independent samples. It is commonly implemented using a Z-test when sample sizes are large enough to justify the normal approximation.

Pooled Proportion

When assuming that the null hypothesis is true (i.e., the proportions are equal), the pooled proportion is computed by combining the successes and total sample sizes from both groups. This combined estimate is then used to calculate a standard error for the difference between the two sample proportions.

Standard Error of the Difference

This concept quantifies the expected variability in the difference between sample proportions if the null hypothesis is true. It is calculated using the pooled proportion along with the individual sample sizes, and it is necessary for standardizing the difference between the sample proportions into a test statistic.

Test Statistic and p-value

The test statistic, typically a Z-score in the case of large samples, measures how far the observed difference in sample proportions is from the expected difference (usually zero under the null hypothesis), relative to the standard error. The p-value is then derived from this statistic and indicates the probability of observing a test statistic at least as extreme as the one computed, assuming the null hypothesis is true. A small p-value suggests that the observed difference is unlikely under the null hypothesis, leading to its rejection in favor of the alternative hypothesis.

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