Question

Timber yield is approximately equal to the volume of a tree, however, this value is difficult to measure without first cutting the tree down. Instead, other variables, such as height and diameter, may be used to predict a tree’s volume and yield. Researchers wanting to understand the relationship between these variables for black cherry trees collected data from 31 such trees in the Allegheny National Forest, Pennsylvania. Height is measured in feet, diameter in inches (at 54 inches above ground), and volume in cubic feet.

Estimate. Std. Error. t value Pr(>/t|)
Intercept. -57.99. 8.64 -6.72 0.00
height. 0.34. 0.13 2.61 0.01
diameter 4.71 0.26 17.82 0.00

Calculate a 95% confidence interval for the coefficient of height. Use df = 28.
A. 0.119 to 0.561
B. 0.074 to 0.606
C. 4.268 to 5.152
D. 4.178 to 5.242

Answer 1

The 95% confidence interval for the coefficient of height is calculated using the estimated coefficient, its standard error, and the critical t-value from the t-distribution with degrees of freedom. The calculated confidence interval is from 0.074 to 0.606.

Step-by-step explanation:

The question asks to calculate a 95% confidence interval for the coefficient of height in a linear regression model that predicts the volume of black cherry trees using variables of height and diameter. Given are the estimated coefficient (0.34), its standard error (0.13), and the degrees of freedom (df = 28).

To find the confidence interval, we use the t-distribution and the provided statistics. The formula for a confidence interval is estimated coefficient ± (t-value) * (standard error). The critical t-value for a 95% confidence interval with 28 degrees of freedom can be found in a t-distribution table, which is approximately 2.048.

Therefore, the confidence interval is calculated as 0.34 ± (2.048 * 0.13), resulting in an interval of 0.074 to 0.606. This is the range where we expect the true population coefficient of height to fall with 95% confidence.