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 $z_{\alpha }$.

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$\left(a\right)-\left(c\right)$.

Short answer

Part$\left(a\right)$ : The$z$-score having an area of $\alpha$to its right under standard normal curve is denoted by $z_{\alpha }$.

Part $\left(b\right)$: The localid="1652700835487" $z$-score having an area of localid="1652700841622" $\alpha$to its left under standard normal curve is denoted by localid="1652700830370" $-z_{\alpha }$.

Part localid="1652700847157" $\left(c\right)$: The two localid="1652700852005" $z$-scores that divide the area under the curve into a middle localid="1652700857606" $1-\alpha$area and two outside areas of localid="1652700862640" $\alpha /2$are localid="1652700867211" $z_{\alpha /2}$and localid="1652700871939" $-z_{\left(\alpha /2\right)}$.

Part localid="1652700877022" $\left(d\right)$: The graphs to illustrate the results in parts localid="1652700881474" $\left(a\right)-\left(c\right)$is,

Step-by-step solution

Part a Step 1. Given information

Let$0<\alpha <1$.

Part a Step 2. Find the required z-score for the given area.

The $z$-score having an area of $\alpha$to its right under standard normal curve is denoted by$z_{\alpha }$.

Part b Step 1. Find the required z-score for the given area.

The$z$-score having an area of$\alpha$to its left under the standard normal curve is denoted by$-z_{\alpha }$.

Part c Step 1. Find the two z-scores for the given areas.

The $z$-score that have an area of $\alpha /2$to its right under standard normal curve is denoted by $z_{\alpha /2}$and the $z$-score that have an area of $\alpha /2$to its left under standard normal curve is, denoted by $-z_{\left(\alpha /2\right)}$.

Part d Step 1. The graphs that illustrate the results in part a-c is,