Regression coefficient confidence interval is a function to calculate the confidence interval, which represents a closed interval around the population regression coefficient of interest using the standard approach and the noncentral approach when the coefficients are consistent.
βj = value of regression coefficient k = number of predictors n = sample size SEβj = standard error α = percentage of confidence interval t = t-Value
The president of a large university wishes to estimate the average age of the students presently enrolled. From past studies, the standard deviation is known to be 6 years. A sample of 40 students is selected, and the coefficient is found to be 0.6 and standard error for the coefficient is 0.25. Find the 95%, 90% and 99% confidence intervals of the population mean.
βj = 0.6, SEβj = 0.25, n = 40, k = 6,
Regression coefficient Confidence Interval (CI)
Calculation of 99% Confidence Interval:
Case 1:Calculate the t value from the given formula, t(1-α/2,n-k-1) α = 99/100 = 0.99 t(1-α/2,n-k-1) = t[(1-0.99)/2,(40-6-1)] = t[0.005,33] = 2.7333
Case 2:Calculation of 99% CI:
Substitute the values in the Confidence Interval formula,
99% CI = 0.6 ± 2.7333 x 0.25
= 0.6 ± 0.683325
= 0.6 + 0.683325, 0.6 - 0.683325
= 1.2833, -0.0833
Calculation of 95% Confidence Interval:
Case 1:Calculate the t value from the given formula, t(1-α/2,n-k-1) α = 95/100 = 0.95 t(1-α/2,n-k-1) = t[(1-0.95)/2,(40-6-1)] = t[0.025,33] = 2.0345
Case 2:Calculation of 95% CI :
Substitute the values in the Confidence Interval formula,
95% CI = 0.6 ± 2.0345 x 0.25
= 0.6 ± 0.508625
= 0.6 + 0.508625, 0.6 - 0.508625
= 1.1086, 0.0914
Calculation of 90% Confidence Interval:
Case 1:Calculate the t value from the given formula, t(1-α/2,n-k-1) α = 0.9 t(1-α/2,n-k-1) = tt[(1-0.9)/2,(40-6-1)] = t[0.05,33] = 1.6924
Case 2:Calculation of 90% CI:
Substitute the values in the Confidence Interval formula,
90% CI = 0.6 ± 1.6924 x 0.25
= 0.6 ± 0.4231
= 0.6 + 0.4231, 0.6 - 0.4231
= 1.0231, 0.1769
