Q. 6.85 - Elementary Statistics

Question

Let $0<\alpha <1$. Determine the

a. $z$-score having an area of $\alpha$ to its right in terms of $z_{\alpha }$.

b. $z-$score having an area of $\alpha$ to its left in terms of $z_a$.

c. Two $z-$scores that divide the area under the curve into a middle $1-\alpha$ area and two outside areas of $\alpha /2$.

d. Draw graphs to illustrate your results in parts (a)-(c).

Step-by-Step Solution

Answer

a. The $z-$score is $z_{\alpha }$

b. The $z-$score is $-z_{\alpha }$

c. $z_{\alpha /2}\text{ and }-z_{a/2}$

d. The graph is:

Part (a) Step 1: Given Information

Calculate the $z-$score having an area of $\alpha$ to its right in terms of $z_a$

Part (a) Step 2: Explanation

We have,

$0<\alpha <1$

Calculation:

$z_{\alpha }$ is the $z-$score under the standard normal curve that has an alpha area to the right.

Part (b) Step 3: Given Information

Calculate the $z-$score having an area of $\alpha$ to its left in terms of $z_{\alpha }$

Part (b) Step 4: Explanation

We have,

$0<\alpha <1$

Calculation:

$z_{\alpha }$ is the $z-$score with an alpha area to its left under the standard normal curve.

Part (c) Step 5: Given Information

Calculate the two $z-$scores that divide the area under the curve into a $1-\alpha$ region in the middle and two $\frac{\alpha }{2}.$ areas on the outside.

Part (c) Step 6: Explanation

We have:

$0<\alpha <1$

Calculation:

There are two $z-$scores that divide the area under the curve into the center ($1-\alpha$) and the two outsides ($\alpha /2$).

It is denoted by the $z-$score $z_{\alpha /2}$ when the $z-$score has an $\alpha /2$ area to its right under the standard normal curve.$-Z_{\alpha /2}$corresponds to the $z-$score that has an area of $\alpha /2$ under the standard normal curve.

Part (d) Step 7: Given Information

Plot the graphs to illustrate your results in parts (a)-(c)

Part (d) Step 8: Final Answer

We have,

$0<\alpha <1$

The following graph depicts the $z-$score with an area of alpha to its right under the standard normal curve:

The following graph depicts the $z-$score with an area of alpha to its left under the standard normal curve:

It is denoted by the score $z_{\alpha /2}$when the $z-$score has an $\alpha /2$ area to its right under the standard normal curve $-Z_{\alpha /2}$ corresponds to the $z-$score that has an area of $\alpha /2$ under the standard normal curve.