Question

T12.3 Inference about the slope ${\beta }_1$ of a least-squares regression line is based on which of
the following distributions?
a. The $t$distribution with $n-1$ degrees of freedom
b. The standard Normal distribution
c. The chi-square distribution with $n-1$ degrees of freedom
d. The $t$ distribution with $n-2$ degrees of freedom
e. The Normal distribution with mean $\mu$ and standard deviation $\sigma$.

Short answer

The correct answer is option (d) The $t$distribution with $n-2$ degrees of freedom.

Step-by-step solution

Given information

To inference about the slope ${\beta }_1$ of a least-squares regression line is based on which of the following distributions.

Explanation

To figure out the below distributions biases inference about a least-squares regression line's slope ${\beta }_1$.
To establish the least square regression line, two parameters must be estimated: the $y$-intercept and the slope.
The $t$-distribution with degrees of freedom equal to the sample size $n$ minus the number of calculated parameters becomes the appropriate distribution.
As a result,
$df=n-2$
Therefore, the correct answer is option (d).