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 \(25 \% . "\) 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 greater for college graduates than it is for those without a high school degree? Test the appropriate hypotheses using a significance level of 0.01 .

Short answer

The final conclusion depends on the calculated p-value. If it is less than the alpha level (0.01), it can be concluded that there's a statistically significant evidence that college graduates have a higher proportion experiencing sunburn compared to the group without a high school degree. If the p-value is higher than the alpha level, there's no enough evidence for this conclusion.

Step-by-step solution

Define Hypotheses

The null hypothesis (\(H_0\)) is that the proportions of college graduates and non-high school graduates getting sunburned are the same. The alternative hypothesis (\(H_A\)) is that the proportion of college graduates getting sunburned is higher. So, \(H_0: p1 = p2\) and \(H_A: p1 > p2\) where \(p1\) is for college graduates and \(p2\) for non-high school graduates.

Compute the Pooled Sample Proportion

The pooled sample proportion (\(p\)) computes the total number of success events (sunburns) over the total sample size. We need the college graduate sunburn rate (0.43), non-high school graduate sunburn rate (0.25) and the total sample size (400) for this. It is calculated as follows: \(p = (p1 * n1 + p2 * n2) / (n1 + n2)\).

Calculate the Standard Error

The standard error (SE) calculates the statistical accuracy of an estimate. It is calculated as follows: \(SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }\).

Compute the z-score

The z-score measures the number of standard deviations an element is from the population mean. It is calculated as follows: \(z = (p1 - p2) / SE\).

Determine the Probability Associated with Observed z-score

Having computed the z-score, it is necessary to calculate the probability or p-value associated with observed z-score. Use a standard normal distribution table or a calculator which offers functions for this.

Draw Conclusion

With the calculated p-value, make a conclusion against the significance level. If the p-value is less than the alpha level (0.01), reject the null hypothesis and conclude that the proportion experiencing sunburn is greater for college graduates. If the p-value is greater than the alpha level, there is not enough evidence to reject the null hypothesis