Chapter 6: Q. 6.8 (page 269)

Let $0 < α < 1$ . Determine the
a. $z$ -score having an area of $α$ to its right in terms of $z_{α}$ .
b. $z$ -score having an area of $α$ to its left in terms of $z_{α}$ .
c. two $z$ -scores that divide the area under the curve into a middle $1 - α$ area and two outside areas of $α / 2$ .
d. Draw graphs to illustrate your results in parts $(a) - (c)$ .

Short Answer

Expert verified

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

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

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

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

Step by step solution

Part a Step 1. Given information

Let $0 < α < 1$ .

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

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

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

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

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

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