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When we want to compare the proportions between two different groups, the two-sample proportion test is our go-to statistical method. In this particular example, we're interested in determining if the proportion of individuals who experience sunburn among college graduates is significantly higher than those without a high school diploma.
To begin, we identify two different samples and calculate the proportion for each group. These proportions represent the percentage of each group experiencing the event of interest—in this case, sunburn. The proportions are expressed as \( p_1 \) and \( p_2 \). In our exercise, \( p_1 = 0.43 \) for college graduates and \( p_2 = 0.25 \) for non-high school graduates.
We use this test to determine if the difference between these two proportions is large enough to reject the null hypothesis, which posits no difference between the groups. The alternative hypothesis, on the other hand, proposes that there is indeed a discernible difference between the two groups.'
Understanding this set up can help clarify not only the results of this particular problem but the process of hypothesis testing more generally.

The significance level, often denoted by \( \alpha \), is a critical component in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true. In simpler terms, it is the threshold for determining whether the results we observe are due to chance or represent a true difference.
For our sunburn study, the significance level is set at 0.05, which is a common choice in many scientific studies. This means we are willing to accept a 5% chance of falsely claiming that college graduates have a distinctly higher incidence of sunburn when, in fact, they do not.
If our test result has a p-value lower than 0.05, we reject the null hypothesis and accept that there is strong evidence to support the alternative hypothesis (i.e., the proportions differ).

• A smaller significance level means more stringent criteria (e.g., 0.01 indicates a 1% risk of error).
• Higher significance levels (e.g., 0.10) reduce the threshold, accepting more risk of error.

This level is determined before conducting the test and influences the interpretation of your outcomes.

The standard error (SE) is a measure of the variability of the sampling distribution of a statistic—in this case, the difference between two sample proportions. Think of it as a way to capture the uncertainty inherent in using sample data to estimate population values.
In the sunburn example, calculating standard error involves first computing the pooled proportion, which aggregates data from both groups. The pooled proportion \( p \) is calculated as follows:\[ p = \frac{p_{1} \times n + p_{2} \times n}{n + n} \]where \( n \) is the sample size for each group, which in this case is 200 for both groups.
Once you have the pooled proportion, the standard error of the difference between the proportions is given by:\[ SE = \sqrt{ p \times ( 1 - p ) \times \left( \frac{1}{n} + \frac{1}{n} \right) } \]

• A larger standard error indicates more variability and thus a less precise estimate of the population parameter.
• A smaller standard error suggests more confidence in the sample statistic as an estimator of the population parameter.

The standard error is essential for calculating the z-score, which determines statistical significance.

The z-score is a crucial part of hypothesis testing, allowing us to determine how far away our observed results are from the null hypothesis. Specifically, the z-score in the context of a two-sample proportion test shows the number of standard errors by which the observed difference between two sample proportions deviates from the null hypothesis of zero difference.
For our sunburn proportions, we calculate the z-score using:\[ Z = \frac{(p_{1} - p_{2})}{SE} \]where \( p_1 = 0.43 \) (college graduates' sunburn proportion), \( p_2 = 0.25 \) (non-high school graduates' sunburn proportion), and \( SE \) is the standard error previously calculated.
Higher absolute values of the z-score indicate a more pronounced deviation from the null hypothesis:

• If the z-score is positive and greater than the critical z-value (like 1.645 for a 5% significance level and a one-sided test), this suggests that the observed proportion difference is significant.
• A lower z-score means the observed differences could easily be due to random sampling variability rather than a true problem.

Z-score calculation ultimately helps us determine whether the evidence supports the hypothesis that a meaningful difference exists between the two groups.