91Ó°ÊÓ

When we perform hypothesis testing in statistics, the null hypothesis, often represented as \( H_0 \) serves as a starting assumption for the test. It is the default position that there is no effect or no difference. In our exercise, the null hypothesis is formulated as \( p_{1} \geq p_{2} \) which states that the proportion of successes in population 1 is at least as large as in population 2. By assuming the null hypothesis is true, we challenge it with sample data to determine if there is enough evidence to reject it in favor of the alternative hypothesis, thereby proving the effect or difference.

Conversely, the alternative hypothesis \( H_1 \) or \( H_a \) represents what we aim to support or prove through our testing. It is the hypothesis that there is a significant effect or a difference between populations or parameters. For the given problem, the alternative hypothesis is formulated as \( p_{1} < p_{2} \), suggesting that the proportion of successes in population 1 is less than that in population 2. This is what we would conclude if we find sufficient evidence against the null hypothesis.

To compare two proportions, as in our exercise, the pooled proportion represents the combined success rate of both samples and is used to evaluate the differences between the proportions. It is calculated by dividing the total number of successes in both samples by the total number of observations. The formula for the pooled proportion is given by:
\( p_{pooled} = \frac{\text{Total successes in both samples}}{\text{Total observations in both samples}} \)
For our exercise, the pooled proportion helped establish a common success rate to compare the individual sample proportions, \( p_{1} \) and \( p_{2} \) under the assumption that the null hypothesis is true.

The test statistic is a calculated value that allows us to decide whether to reject the null hypothesis. It is determined using sample data and reflects how far the observed data deviates from the null hypothesis. In the case of proportion tests, the test statistic often follows a Z-distribution. For this exercise, we used the following formula to calculate the Z test statistic:
\( Z = \frac{(p_{1} - p_{2})}{\sqrt{p_{pooled}(1 - p_{pooled})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}} \)
After substituting the sample and pooled proportions, this statistic is then compared to a critical value to decide whether the evidence is strong enough to reject \( H_0 \) in favor of \( H_1 \. In our exercise, the calculated Z-value did not fall beyond the critical value; thus, we did not reject the null hypothesis.

The significance level, denoted as \( \alpha \) in hypothesis testing, is a threshold for acceptable error while making a decision based on the test result. It's the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. Common \( \alpha \) levels are 0.05, 0.01, and 0.10. The exercise used a significance level of 0.10, indicating we're willing to accept a 10% chance of mistakenly rejecting the true null hypothesis. The critical value corresponding to \( \alpha \) for a one-tailed test is used to determine the rejection region for the null hypothesis. If the test statistic falls into that region, we would reject \( H_0 \- otherwise, we fail to reject it.