Chapter 13.3, Problem 29PB

Mental health again Refer to the previous exercise.

1. a. Report the test statistic and P-value for testing H₀: β₁ = β₂ = 0.
2. b. State the alternative hypothesis that is supported by the result in part a.
3. c. Does the result in part a imply that necessarily both life events and SES are needed in the model? Explain.

13.28 Regression for mental health A study in Alachua County, Florida, investigated an index of mental health impairment, which had $\bar{y}$ = 27.3 and s = 5.5. Two explanatory variables were x₁ = life events score (mean = 44.4, s = 22.6) and x₂ = SES (socioeconomic status, mean = 56.6, s = 25.3). Life events is a composite measure of the number and severity of major life events, such as death in the family, that the subject experienced within the past three years. SES is a composite index based on occupation, income, and education. The table shows data for 6 subjects. The complete data¹ for a random sample of 40 adults in the county is in the Mental Health data file on the book’s website. Some regression results are also shown.

1. a. Find the 95% confidence interval for β₁.
2. b. Explain why the interval in part a means that an increase of 100 units in life events corresponds to anywhere from a 4- to 17-unit increase in mean mental impairment, controlling for SES. (This lack of precision reflects the small sample size.)

y = mental impairment, x₁ = life events index, and x₂ = socioeconomic status