# Learn How to Calculate Regression Coefficient Confidence Interval – Tutorial

## Calculate Regression Coefficient Confidence Interval - Definition, Formula and Example

##### Definition:

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.

#### Formula: ###### Where,

βj = value of regression coefficient k = number of predictors n = sample size SEβj = standard error α = percentage of confidence interval t = t-Value

##### Example :

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.

##### Given,

βj = 0.6, SEβj = 0.25, n = 40, k = 6,

##### To Find,

Regression coefficient Confidence Interval (CI)

#### Solution :

##### Step 1:

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

##### Step 2:

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

##### Step 3:

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