Question

"Smartest People Often Dumbest About Sunburns" is the headline of an article that appeared in the San Luis Obispo Tribune (July 19,2006 ). The article states that "those with a college degree reported a higher incidence of sunburn that those without a high school degree- \(43 \%\) versus \(25 \%\)." For purposes of this exercise, suppose that these percentages were based on random samples of size 200 from each of the two groups of interest (college graduates and those without a high school degree). Is there convincing evidence that the proportion experiencing a sunburn is higher for college graduates than it is for those without a high school degree? Answer based on a test with a .05 significance level.

Short answer

The short answer to the exercise will depend on the calculated Z-value. If the Z-value is greater than 1.645, the answer would be 'Yes, there is convincing evidence at a 0.05 significance level that the proportion experiencing a sunburn is higher for college graduates than it is for those without a high school degree'. If the Z-value is less than or equal to 1.645, the answer would be 'No, there is not convincing evidence at a 0.05 significance level that the proportion experiencing a sunburn is higher for college graduates than it is for those without a high school degree'.

Step-by-step solution

- Setup the Hypotheses

The null hypothesis (H0) indicates no difference between the two population proportions while the alternative hypothesis (H1) indicates a higher proportion among the college graduates. Therefore: \n\nH0: p1 - p2 = 0 (where p1 is the proportion of college graduates who experience sunburn and p2 is the proportion of non-high school graduates who experience sunburn) \n\nH1: p1 - p2 > 0 (indicating a higher proportion of sunburn among college graduates)

- Calculate the Sample Proportions

Calculate the sample proportions using the percentages provided: \n\np1 = \(43 \% = 0.43\) (college graduates) \np2 = \(25 \% = 0.25\) (non-high school graduates) \nAlso, it is told that each group has a size of 200.

- Calculate the Test Statistic

The test statistic (Z) can be calculated using the formula: \n\nZ = \(\frac {(p1 - p2) - 0} {\sqrt {\frac {(p1(1-p1))} {n1} + \frac {(p2(1-p2))} {n2}}} \), where n1 and n2 are the sample sizes. Plugging in the numbers from the given data we get the Z value.

- Determine the Critical Value and Decision Rule

For a level of significance (α) of 0.05 and a one-tailed test, the critical value from the Z table is approximately 1.645. If the resulting test statistic is greater than 1.645, the null hypothesis will be rejected in favor of the alternative hypothesis, indicating a significant difference in sunburn rates between the two groups.

- Make a Decision and Interpret the Result

After calculating Z, compare it to the critical value. If Z > 1.645, reject the null hypothesis and conclude that there is evidence to suggest that the proportion of college graduates experiencing sunburn is higher than the proportion of non-high school graduates experiencing sunburn. If Z ≤ 1.645, fail to reject the null hypothesis and conclude that there is no significant difference in sunburn rates among the two populations.

Key concepts

Null Hypothesis

The null hypothesis is a foundational concept in hypothesis testing. It serves as the default or starting assumption, proposing that there is no effect or no difference between two compared groups. In the context of our sunburn study, the null hypothesis asserts that the incidence of sunburn is the same for both college graduates and non-high school graduates.

Formally, the null hypothesis is denoted as H0: p1 - p2 = 0, where p1 and p2 represent the population proportions of the two groups in question. If our test concludes that the data significantly deviates from this hypothesis, we may have grounds to reject it. Otherwise, we lack sufficient evidence and therefore retain it, maintaining that there's no significant difference.

Alternative Hypothesis

Conversely, the alternative hypothesis is the opposite of the null hypothesis, indicating that there is an effect or a difference between the groups being compared. In our sunburn example, the alternative hypothesis posits that college graduates have a higher incidence of sunburn compared to those without a high school degree.

The alternative hypothesis can be expressed as H1: p1 - p2 > 0, suggesting that the proportion p1 is greater than p2. It is what researchers are typically trying to gather evidence for, as finding support for the alternative hypothesis can indicate a need for a change in belief or policy.

Test Statistic

The test statistic is a calculated value that we compare to a critical value in order to make a decision about the hypotheses. It quantifies the difference between the observed sample result and what we would expect under the null hypothesis, standardized by the variability in the data.

For the sunburn study, the test statistic is a Z-score given by Z = \(\frac{(p1 - p2) - 0}{\sqrt{\frac{p1(1-p1)}{n1} + \frac{p2(1-p2)}{n2}}}\) where p1 and p2 are the sample proportions and n1 and n2 are the sizes of the respective samples. A larger absolute value of the test statistic indicates a larger difference between the groups and provides stronger evidence against the null hypothesis.

Level of Significance

The level of significance, often denoted by α (alpha), is the probability threshold at which we reject the null hypothesis. It reflects the risk we are willing to take of incorrectly rejecting the null hypothesis when it is actually true, an error known as a Type I error.

In our sunburn study, a .05 significance level means there’s a 5% chance we would reject the null hypothesis due to random sampling error when there is actually no difference in the population. The critical value associated with our chosen level of significance helps us to make the decision: if our test statistic exceeds this value, we reject the null hypothesis.

Sample Proportions

Sample proportions are estimates of the population proportions that we derive from our data. They provide insight into the characteristics of the population from which the sample was drawn. In our exercise, the calculated sample proportions are p1 = 43% and p2 = 25% for college graduates and non-high school graduates, respectively.

These sample proportions are essential in assessing the variability of the outcomes and are used to compute the test statistic. It's crucial to use a random sample, as it allows the sample proportions to be more representative of the population, More accurate sample proportions mean a more reliable test statistic and hence, a more trustworthy hypothesis test.