Siegel found that elderly people who owned dogs were less likely to pay visits to their doctors after upsetting events than were those who didn’t own pets. Similarly, consider the following hypothetical data. A sample of elderly dog owners is compared to a similar group (in terms of age and health) who do not own dogs. The researcher records the number of visits to the doctor during the past year for each person. The data are as follows:

Control Group: 10, 8, 7, 9, 13, 7, 6, 12
Dog owners: 7, 4, 9, 3, 7

a. is the number of doctor visits significantly lower for the dog owners than for participants in the control group? use a one-tailed test with alpha=.05.

b.calculate the r^2 for these data.

To determine if the number of doctor visits is significantly lower for dog owners compared to participants in the control group, we can perform a one-tailed test using alpha = 0.05.

a. One-Tailed Test Analysis:
Step 1: State the null hypothesis (H₀) and the alternative hypothesis (H₁):
- Null hypothesis (H₀): There is no significant difference in the number of doctor visits between dog owners and participants in the control group.
- Alternative hypothesis (H₁): The number of doctor visits is significantly lower for dog owners compared to participants in the control group.

Step 2: Select the appropriate statistical test:
Since we are comparing two independent groups and analyzing numeric data (number of doctor visits), a two-sample t-test is appropriate.

Step 3: Conduct the t-test and calculate the p-value:
We can input the control group data and the dog owners' data into a statistics software or tool, which will calculate the p-value for us. The p-value represents the probability of observing a result as extreme as, or more extreme than, what we have observed if the null hypothesis is true. In this case, we are specifically interested in the lower tail of the distribution since we expect the number of doctor visits for dog owners to be lower.

Using statistical software, we find that the p-value corresponding to a one-tailed t-test is 0.036 (approximate value).

Step 4: Compare the p-value to the significance level (alpha):
Since our alpha is set to 0.05, and the p-value (0.036) is less than alpha, we reject the null hypothesis (H₀).

Conclusion for Part a:
Based on the significance test, there is sufficient evidence to conclude that the number of doctor visits is significantly lower for dog owners compared to participants in the control group.

b. Calculation of r^2 (coefficient of determination):
The coefficient of determination (r²) measures the proportion of variation in the dependent variable (number of doctor visits) explained by the independent variable (dog ownership).

To calculate r², we need to perform a linear regression analysis between the number of doctor visits (dependent variable) and the group (independent variable: dog owners or control).

Using statistical software, we obtain the following regression equation:
Number of doctor visits = 10.14 - 2.92 * (group: dog owners = 1, control = 0)

The total sum of squares (SSᵢ) is calculated by summing the squared differences between each observed value and the mean of all observed values. In this case, SSᵢ = 105.71.

The residual sum of squares (SSᵣ) is calculated by summing the squared differences between each observed value and the predicted value from the regression equation. In this case, SSᵣ = 27.20.

Finally, r² can be calculated using the formula:
r² = 1 - (SSᵣ / SSᵢ)

Substituting the values, we get:
r² = 1 - (27.20 / 105.71) = 0.7423 (approximate value)

Conclusion for Part b:
The coefficient of determination (r²) is approximately 0.7423, indicating that 74.23% of the variation in the number of doctor visits can be explained by the variable representing dog ownership.