Question

Physical education requirements. In Exercise 8.41 (page 482), you analyzed data from a study that included 354 higher education institutions: 225 private and 129 public. Among the private institutions, 60 required a physical education course, while among the public institutions, 101 required a course. Your analysis in that exercise focused on the comparison of two proportions. Use these data to construct a two-way table for analysis and find the joint distribution, the marginal distributions, and the conditional distributions. Use these distributions to give a brief summary of the relationship between the type of institution and whether a physical education course is required. 9.8 Significance test for physical education requirements. Refer to Exercise 9.2. Find the chi-square statistic, the degrees of freedom, and the P-value. What do you conclude?

Answer 1

To summarize the question, we create a two-way table based on the data provided, determine the distributions, and use those to perform a chi-square test to identify any relationship between the type of institution and the requirement of a physical education course.

Step-by-step explanation:

To analyze the relationship between the type of institution and the requirement of a physical education course using the given data, we construct a two-way table and calculate the joint, marginal, and conditional distributions. With 225 private institutions, 60 requiring physical education, and 129 public institutions, 101 requiring physical education, the two-way table would look like this:

Private Institutions Requiring PE: 60

Private Institutions Not Requiring PE: 165

Public Institutions Requiring PE: 101

Public Institutions Not Requiring PE: 28

The joint distribution is the probability of both events happening together, while marginal distribution is the probability of a single event regardless of the other event. The conditional distribution shows the probability of one event given that the other event has occurred.

To find the chi-square statistic, degrees of freedom, and the p-value, you could use the data to perform a chi-square test for independence. The chi-square statistic will indicate how likely it is that any observed difference between the sets of numbers is due to chance. Degrees of freedom, typically calculated as (rows-1)*(columns-1) for a two-way table, would be used in determining the p-value from the chi-square distribution. You would compare the p-value to a significance level to decide whether to reject the null hypothesis, which states there is no association between the type of institution and the requirement of a physical education course.

Type I and Type II errors relate to incorrect rejection or non-rejection of this null hypothesis: a Type I error would occur if we wrongly conclude there is a difference when there is not, and a Type II error occurs if we fail to detect a difference that truly exists.

Answer 2

A two-way table is created to visualize the relationship between institution types and physical education requirements. Joint, marginal, and conditional distributions are calculated to understand the proportions and associations. A chi-square test for independence is used to assess the statistical significance of the relationship.

Step-by-step explanation:

To address the student's question about physical education requirements using the provided data, we begin by constructing a two-way table. This table will help visualize the relationship between institution types and the requirement of a physical education course. After constructing the table, we then calculate the joint distribution, which gives us the proportion of each combination of institution type and physical education requirement in the whole dataset. Next, we determine the marginal distributions, which show the proportion of each institution type and the proportion requiring physical education independent of institution type. The conditional distributions tell us the proportion of institutions requiring physical education given that they are either private or public.

After we have calculated these distributions, we can summarize the relationship between the type of institution and the requirement for a physical education course. As the number of private institutions requiring a physical education course is lower than that of public institutions, we see an association. Following this, we can conduct a chi-square test for independence to determine if the observed relationship is statistically significant. We calculate the chi-square statistic, degrees of freedom, and P-value to assess this. If our P-value is below a chosen significance level, such as 0.05 or 0.01, we can reject the null hypothesis, suggesting there is a statistically significant difference in physical education requirements between private and public institutions.