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).

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

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.

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

We have,

$0<\alpha <1$

Calculation:

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

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.

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.

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

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.