Chapter 13, Problem 1CQ

A confidence interval for ${\beta }_1$ is to be constructed from a sample of 20 points. How many degrees of freedom are there for the critical value?

To determine

To find:the number of degrees of freedom for the critical value

Answer to Problem 1CQ

Number of degrees of freedom is $18$

Explanation of Solution

Given information: A confidence interval for ${\beta }_1$ to be constructed from a sample of $20$ points.

the steps to be followed when constructing a confidence interval for ${\beta }_1$ .

Step $1$ : compute the least square regression line $\stackrel{\wedge }{y}=b_0+b_{1}x$

Step $2$ : compute the residuals and construct a residual plot $y-\stackrel{\wedge }{y}$ to be sure the assumptions of the linear model are satisfied.

Step $3$ : compute the residual standard deviation $s_e=\sqrt{\frac{(y-\stackrel{\wedge }{y{)}^2}}{n-2}}$

Step $4$ : compute the standard error of $b_1:s_b=\frac{s_e}{\sqrt{{\displaystyle \sum (x-\overline{x}}}{)}^2}$

Step $5$ : Find the critical value $t_{\alpha /2}$ from the student’s $t$ curve with $(n-2)$ degrees of freedom and multiply it by the standard error to obtain the margin of error $t_{\alpha /2}.s_b$

Step $6$ : Use the point estimate $b_1$ and the margin of error $t_{\alpha /2}.s_b$ to compute the confidence interval.

Point estimate $\pm$ margin of error

Step $7$ : Interpret the result

Therefore, as mentioned in the Step $5$$(n-2)$ = $20-2=18$ number of degrees of freedom for a critical value $t_{\alpha /2}$ .