A study in Alachua County, Florida, investigated an index of mental health impairment, which had 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 are a composite measure of the number and severity of major life events, such as a 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 data 1 for a random sample of 40 adults in the county is in the Mental Health data file on the text CD. Some regression results are also shown.
a. Find the 95% confidence interval for β1.
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.)
a. b1 + t.025 (se) = 0.10326 + 2.026(0.03250) = 0.10326 ± 0.06585, or (0.04, 0.17).
b. The confidence interval gives plausible values for the slope, the amount that means mental impairment will increase when life events score increases by one when controlling for SES. For an increase in 100 units of life events, we multiply each endpoint of the confidence interval by 100, giving (4, 17); this indicates that plausible increases in mean mental impairment range from 4 to 17 when life events scores increase by 100, controlling for SES.