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?

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀: The proportion of 18- to 24-year-olds who benefit mentally from taking classes in parks is equal to or greater than the proportion who benefit mentally from enjoying nature in parks. H₁: 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.
2. Step 2: Identify the sample proportions from the data provided in the image. The proportion for enjoying nature is 44% (p₁ = 0.44), and the proportion for taking classes is 39% (p₂ = 0.39). Also, note the sample size (n₁ = n₂ = 200).
3. Step 3: Calculate the test statistic for comparing two proportions. Use the formula: z = (p₁ - p₂) / sqrt(p̂(1 - p̂)(1/n₁ + 1/n₂)), where p̂ = (x₁ + x₂) / (n₁ + n₂). Here, x₁ and x₂ are the number of successes for each group, and n₁ and n₂ are the sample sizes.
4. Step 4: Determine the critical value for α = 0.05 in a one-tailed test. Look up the z-value corresponding to a significance level of 0.05 in a z-table. This will be the threshold for rejecting the null hypothesis.
5. Step 5: Compare the calculated test statistic to the critical value. If the test statistic is less than the critical value, reject the null hypothesis and conclude 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. Otherwise, fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1). In this context, the null hypothesis would state that the proportion of 18- to 24-year-olds benefiting from taking classes is equal to or greater than those benefiting from enjoying nature, while the alternative hypothesis would claim the opposite.

Proportions

Proportions are a way to express a part of a whole, often represented as a fraction or percentage. In this question, we are comparing the proportions of two groups: those who benefit from taking classes (39%) and those who benefit from enjoying nature (44%). Understanding how to calculate and interpret these proportions is crucial for analyzing the survey data and drawing conclusions.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis in hypothesis testing. A common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In this scenario, using α=0.05 means that if the p-value from the test is less than 0.05, we can conclude that there is significant evidence to support the claim that fewer young adults benefit from taking classes compared to enjoying nature.