Question

A sociologist wishes to study the relationship between happiness and age. He interviews 24 individuals and collects data on age and happiness, measured on a scale from 0 to 100. He estimates the following model: Happiness = β₀ + β₁Age + ε. The following table summarizes a portion of the regression results.

	Coefficients	Standard Error	t-stat	p-value
Intercept	56.184	5.2123	10.7791	0.0000
Age	0.2811	0.0887	3.1691	0.0023

At the 5% significance level, which of the following is the correct confidence interval of the regression coefficient β₁?

a [0.0971, 0.1111]

b [0.0771, 0.4651]

c [0.0771, 0.1111]

d [0.0971, 0.4651]

Answer 1

From given output we get,

Estimate for the slope is b1 = 0.2811

Standard error for the slope = SE = 0.0887

sample size (n) = 24

Degrees of freedom = 24 - 2 = 22

Using t-table we get, t-critical value at significance level of 0.05 with 22 degrees of freedom is, tc = 2.074

The 95% confidence interval for the regression coefficient β1 is,

b1 - (tc * SE) < β1 < b1 + (tc * SE)

0.2811 - (2.074 * 0.0887) < β1 < 0.2811 + (2.074 * 0.0887)

0.2811 - 0.1840 < β1 < 0.2811 + 0.1840

0.0971 < β1 < 0.4651

Answer : d) [ 0.0971, 0.4651 ]