Random samples of employees were drawn in fastfood...

Random samples of employees were drawn in fastfood restaurants where the employer provides a training program. Of a sample of 67 employees who had not completed high school, 11 had participated in a training program provided by their current employer. Of an independent random sample of 113 employees who had completed high school but had not attended college, 27 had participated. Test, at the $1 \%$ level, the null hypothesis that the participation rates are the same for the two groups against the alternative that the rate is lower for those who have not completed high school.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis, which is assumed to be true, and an alternative hypothesis, which is what you want to test for. The decision is based on calculating a test statistic and comparing it to a threshold (or computing a p-value) at a specified significance level.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement that there is no effect or difference—in this case, that the participation rates in the training program are the same for employees with and without a high school education. The alternative hypothesis (H?) is the claim to be assessed by the test, here stating that the participation rate for those who have not completed high school is lower. Setting up these hypotheses correctly is crucial to direct the statistical test appropriately.

Two-Sample Test for Proportions

A two-sample test for proportions is used to compare the proportions of a particular outcome between two independent groups. This method determines whether any observed difference in sample proportions is statistically significant or could be attributed to sampling variability. It involves calculating a pooled proportion to estimate the common population proportion under the null hypothesis and then standardizing the difference between groups to form a test statistic.

Significance Level

The significance level, often denoted by alpha (?), is the probability threshold below which the p-value leads to the rejection of the null hypothesis. In this context, a 1% significance level means that there is a 1% risk of concluding that there is a difference when in reality the difference is due to random chance. This stringent level reduces the likelihood of a Type I error, which is falsely rejecting the null hypothesis.

P-value and Test Statistic

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the value observed, assuming the null hypothesis is true. The test statistic, derived from the difference between sample proportions and the pooled estimate, quantifies the deviation from the null hypothesis. Comparing the p-value to the significance level allows the researcher to decide whether to reject or fail to reject the null hypothesis.

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