9.5 Carrying Out a Test for the Slope of a Regression Model

Complete Slope t-Test from Computer Output

Problem: A researcher studies the relationship between hours of sleep (${x}$) and reaction time (${y}$, in ms) for ${n = 20}$ subjects. Computer output shows:

b = -12.3, SE_b = 2.87, T = -4.29, P = 0.000, n = 20

Test whether there is a negative linear relationship at ${\alpha = 0.05}$.

P: ${\beta}$ = true slope of the population regression line relating sleep hours to reaction time

H: ${H_0: \beta = 0}$ vs. ${H_a: \beta < 0}$ (hypothesize less sleep = higher reaction time)

A (LINE):

N: t-test for the slope, ${df = 20 - 2 = 18}$

T: ${t = -12.3/2.87 \approx -4.29}$ (from output)

O: ${p = P(t_{18} \leq -4.29) \approx 0.000/2 \approx 0.000}$ (left-tailed)

M/S: Since ${p < 0.05}$, reject ${H_0}$. We have convincing evidence of a negative linear relationship between sleep hours and reaction time.