Question

14.46 Consider the standardized variable
$z=\frac{b_1-{\beta }_1}{\sigma /\sqrt{S_u}}$
a. Identify its distribution.
b. Why can't it be used as the test statistic for a hypothesis test concerning ${\beta }_1$ ?
c. What statistic is used? What is the distribution of that statistic?

Short answer

(a) The slopes of all possible sample regression lines have a normal distribution.
(b) The value of $\sigma$ is unknown.
(c) The $t$-distribution have standard error of the estimate.

Step-by-step solution

Part (a) Step 1: Given information

To Identify its distribution of $z=\frac{b_1-{\beta }_1}{\sigma /\sqrt{S_u}}$

Part (a) Step 2: Explanation

The given value of the standardized variable as follows:

$z=\frac{b_1-{\beta }_1}{\sigma /\sqrt{S_u}}$

Since its mean is ${\beta }_1$and its standard deviation is $\frac{\sigma }{S_{xx}}$, the distribution of slopes of all feasible sample regression lines.

The slopes of all possible sample regression lines have a normal distribution.

Part (b) Step 1: Given information

To find that the value $z=\frac{b_1-{\beta }_1}{\sigma /\sqrt{S_u}}$ can't it be used as the test statistic for a hypothesis test concerning.

Part (b) Step 2: Explanation

The given value of the standardized variable as follows:

$z=\frac{b_1-{\beta }_1}{\sigma /\sqrt{S_u}}$
Here the value of $\sigma$ is unknown.

Part (c) Step 1: Given information

To explain the statistic is used and the distribution of the statistic.

Part (c) Step 2: Explanation

The given value of the standardized variable as follows:

$z=\frac{b_1-{\beta }_1}{\sigma /\sqrt{S_u}}$

The estimate's standard error, which has a $t$-distribution, is used.
As a result, the estimate's standard error, or $t$-distribution