Parks and Mental Health In Exercises 13–18, use the figure, wh...

Question

Parks and Mental Health In Exercises 13–18, use the figure, which shows the percentages from a survey of two hundred 18- to 24-year-olds in the United States who say that various park and recreation activities have a positive impact on their mental health. (Adapted from National Recreation and Park Association)

Taking Classes and Enjoying Nature At α=0.05, can you support the claim that the proportion of 18- to 24-year-olds who benefit mentally from taking classes in parks is less than the proportion who benefit mentally from enjoying nature in parks?

Accepted Answer

All right, hello, everyone. So, this question says, in a health survey of newly vaccinated individuals, researchers investigated whether there's a difference in the proportion of people aged 18 to 29, and those aged 30 to 49, who received at least one dose of a new vaccine. The study included random samples of 1000 individuals from each age group. Among the 18 to 29, each group, 720 had received at least one dose. Among the 30 to 49 age group, 680 had received at least one dose. At alpha equals 0.01, can you support the claim that the proportion of vaccinated individuals in the 18 to 29 age group is greater than the proportion in the 30 to 49 age group? And here we have 4 different answer choices labeled A through D. All right, so first, what are the known alternative hypotheses for this experiment, for this context? The null hypothesis Hno would state that P1 is equal to P2. In this case, that's the proportion of people in each age group that have received at least one dose of a vaccine. On the other hand, The alternative hypothesis, HA would state that P1 is greater than P2. Because this follows the claim that the 18 to 29 age group has received more vaccines than the second age group. So because of this, this is a right tail test. So now, let's calculate Phat, which represents the pooled proportion across both samples. The pulled proportion is equal to the sum of. Oops. The sum of X1 and X2, which represents the number of people in each group that got vaccinated. Divided by the sum of 1 and 2. So Phat is equal to 720. Added to 680. Divided by 1000 added to 1000. This ultimately equals 0.70. And now we can calculate the standard error based on this. So SE is equal to the square root of. Piha Multiplied by one subtracted by a pea hat. And multiplied by one subtracted, excuse me, one divided by N1. Added to 1 divided by N2. So ultimately, this equals the square root of 0.70. Multiplied by 1 subtracted by 0.70. Multiplied by 1 divided by 1000, added 21 divided by 1000. And after evaluating this expression, SE is approximately equal to 0.02049. Which now brings you to your test statistic. So our Z value is equal to Phat1 subtracted by Phat 2. Divided by SE Now, recall that Phat 1 and Phat 2. Represents the proportion in each individual age group that received vaccines. So, Phat for each sample would be equal to the number of people that got vaccinated divided by 1000, which is the sample size. So, when you calculate this, Phat 1 is equal to 0.72, whereas PAT 2 is equal to 0.68. So Z is equal to 0.72, subtracted by 0.68. Divided by 0.02049. So Z is ultimately equal to 1.952. So this is our test statistic. Now our critical value. Add alpha equals 0.01 for a right-tailed test. Is equal to 2.33. And now, we can see that our test statistic 1.952, is less than our critical value, 2.33. And because this is a right-tailed test, the fact that our test statistic is less than a critical value means that we failed to reject. Are no hypothesis. So ultimately Our correct answer after I scroll up so we can see them. Is option D in the multiple choice, which reads there is insufficient evidence to support that 18 to 29 year olds are more likely to be vaccinated than 30 to 49-year-olds. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Hypothesis Testing

Proportions

Significance Level (α)

Watch next