Q. 6.8 - 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 $z_{\alpha }$.

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 $\left(a\right)-\left(c\right)$.

Step-by-Step Solution

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

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

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

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,