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A null hypothesis is a statement that suggests no effect or no difference between populations. It is the starting point for hypothesis testing, where we aim to determine if there is enough statistical evidence to reject this assumption. In hypothesis testing, researchers often assume the null hypothesis is true until proven otherwise. This forms the basis for evaluating the observed data. In the exercise, the null hypothesis is stated as \(H_0: \pi_1 = \pi_2\), meaning there is no difference between the two population proportions. This assumption simplifies analysis by providing a specific frame of reference.

The alternate hypothesis (or alternative hypothesis) is the statement that contradicts the null hypothesis. It represents what the researcher aims to support or prove through data analysis. It's often denoted as \(H_1\), and in the given exercise, it is \(H_1: \pi_1 eq \pi_2\). This indicates a belief that there is a difference between the two population proportions. When conducting a hypothesis test, if the evidence strongly suggests the null hypothesis is unlikely, the conclusion is to reject the null hypothesis in favor of the alternate hypothesis. This forms the basis for the decision-making process in hypothesis testing.

A z-test is a statistical test used to determine if there is a significant difference between the means or proportions of two groups. It utilizes the z-distribution, which is a special case of the normal distribution, typically when sample sizes are large. Z-tests are useful for testing the hypothesis about population proportions where the sample size is sufficient for the normal approximation of the binomial distribution. In a z-test for proportions, the test statistic is calculated to compare an observed sample proportion with an expected one under the null hypothesis. In our exercise, the calculated z-statistic was approximately 3.195. It was evaluated against the critical value for a two-tailed test at a 0.05 significance level, which is ±1.96. Since the calculated statistic exceeds this threshold, it suggests that the observed difference is unlikely to have occurred by random chance.

Pooled proportion refers to the combined proportion from two samples. It is a useful measure when comparing two population proportions, as it provides a way to standardize these proportions into a single value. The pooled proportion is calculated using the formula:

• \( \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} \)

where \(x_1\) and \(x_2\) are the number of successes in the two samples, and \(n_1\) and \(n_2\) are their respective sample sizes. In the exercise, the pooled proportion was calculated as approximately 0.8. It is important because it allows for the estimation of the standard error and forms part of the z-test calculation. By providing an expected proportion under the null hypothesis, it helps to identify whether the observed differences between sample proportions are statistically significant.