Question

Standard Normal areas Find the proportion of observations in a standard Normal distribution that satisfies each of the following statements.

a. $z>-1.66$

b.$-1.66<z<2.85$

Short answer

Part (a) Around $0.9515$of the observations in a standard Normal distribution satisfy the statement.

Part (b) Around $0.9493$ of the observations in a standard Normal distribution satisfy the statement.

Step-by-step solution

Part (a) Step 1: Given information

$z>-1.66$

Part (a) Step 2: Concept

The formula used: Use the complement rule$P(notA)=1-P\left(A\right)$

Part (a) Step 3: Calculation

To find the equivalent probability, use the normal probability table in the appendix.

In the typical normal probability table for $P(z<-1.66)$look at the row that starts with$-1.6$ and the column that starts with$.06$

Then

Use the complement rule,

$P(notA)=1-P\left(A\right)$

For probability

$$\begin{gathered} P(z>-1.66)P(z>-1.66) \\ =1-P(z<-1.66) \\ =1-0.0485 \\ =0.9515 \end{gathered}$$

Therefore,

Around $0.9515$ of the observations in a standard Normal distribution satisfy$z>-1.66$

Part (b) Step 1: Given information

$-1.66<z<2.85$

Part (b) Step 2: Calculation

To find the equivalent probability, use the normal probability table in the appendix.

In the typical normal probability table for $P(z<-1.66)$look at the row that starts with $-1.6$and the column that starts with$.06$

And

The usual normal probability table for $P(z<2.85)$has a row that starts with $2.8$and a column that starts with$.05$

Then

$$\begin{gathered} P(-1.66<z<2.85) \\ =P(z<2.85)-P(z<-1.66) \\ =0.9978-0.0485 \\ =0.9493 \end{gathered}$$

Therefore,

Around $0.9493$of the observations in a standard Normal distribution satisfy $-1.66<z<2.85$