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A proportion test is a statistical tool used to determine if there is a significant difference between two group proportions. It helps in comparing scenarios, like whether college graduates have a higher rate of sunburn than those without a high school degree. In this scenario, we refer to these groups' proportions as \(p_1\) and \(p_2\).
To start, you identify the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)).

• \(H_0: p_1 = p_2\) indicates no difference between the proportions.
• \(H_a: p_1 > p_2\) suggests that the first group has a higher proportion, meaning more college graduates get sunburnt.

A proportion test checks if we can statistically support the alternative hypothesis. Here, it can affirm whether education level corresponds to sunburn rates.

The significance level in hypothesis testing determines how confident we are about the test results. It's often denoted by alpha (\(\alpha\)), and it defines the probability threshold for rejecting the null hypothesis. In our scenario, the significance level is set at 0.05, or 5%.Choosing a significance level of 0.05 sets a threshold for deciding: if the test results demonstrate a discrepancy between the groups that has only a 5% probability of being due to random chance, we conclude there is a true difference.

• A 5% level indicates a willingness to accept a 5% risk of incorrectly rejecting the null hypothesis.
• This is often a standard in many research fields to balance between being too lenient or too strict.

Hence, observed data need to be significantly beyond expected results under the null hypothesis to make a case for the alternative hypothesis.

Calculating the test statistic is crucial in hypothesis testing, as it measures the extent of the observed difference between group proportions. It involves a few steps:1. **Compute the pooled proportion:** The pooled proportion combines the two sample proportions into a single reference proportion. It is calculated by dividing the total number of individuals with the trait by the total sample size. In our problem, it helps in adjusting the standard error calculation.2. **Determine the standard error:** This reflects how much variation we expect in the proportion differences if the null hypothesis is true. A small standard error suggests the proportions are reliably distinct between samples.3. **Compute the Z-score:** Using the formula \(Z = \frac{{p_1 - p_2}}{{SE}}\), where SE represents the standard error. It signifies the number of standard deviations the observed difference is from the expected difference under the null hypothesis. A higher Z-score points to stronger evidence against \(H_0\).With the calculated Z = 5.0, which significantly surpasses the critical value of 1.645 for a 5% significance level, rejecting the null hypothesis becomes justifiable, confirming a genuine difference in sunburn rates between the education levels.