Chapter 6, Problem 1PS

Given a standardized normal distribution (with a mean of 0 and standard deviation of 1, as in Table E.2), What is probability that

$\begin{array}{l}\text{a}\text{.}Z\text{is}\text{less}\text{than}1.57?\\ \text{b}\text{.}Z\text{is}\text{greater}\text{than}1.84?\\ \text{c}\text{.}Z\text{is}\text{between}1.57\text{and}1.84?\\ \text{d}\text{.}Z\text{is}\text{less}\text{than}1.57\text{greater}\text{than}1.84?\end{array}$

To determine

The probability that $Z$ is less than 1.57.

Answer to Problem 1PS

The required probability is 0.9417.

Explanation of Solution

The Table E.2 given in the Appendix of the book provides the area under the cumulative standardized normal distribution from $-\infty$ to $Z$ .

From the table, the cumulative probability corresponding to the value 1.57 is 0.9418, which can be written as:

$P\left(Z<1.57\right)=0.9417$

Thus, the probability is 0.9417.

To determine

The probability that $Z$ is greater than 1.84.

Answer to Problem 1PS

The required probability is 0.0329.

Explanation of Solution

The probability that $Z$ greater than 1.84 can be written as:

$P\left(Z>1.84\right)=1-P\left(Z<1.84\right)$

From the table E.2, the cumulative probability corresponding to the value 1.84 is 0.9671, which can be written as, $P\left(Z<1.84\right)=0.9671$ .

Substituting the value in the above equation, the required probability is obtained as:

$\begin{array}{c}P\left(Z>1.84\right)=1-0.9671\\ =0.0329\end{array}$

Thus, the probability is 0.0329.

To determine

The probability that $Z$ is between 1.57 and 1.84.

Answer to Problem 1PS

The required probability is 0.0254.

Explanation of Solution

The probability that $Z$ is between 1.57 and 1.84 can be written as:

$P\left(1.57<Z<1.84\right)=P\left(Z<1.84\right)-P\left(Z<1.57\right)$

From the table E.2, $P\left(Z<1.57\right)=0.9417$ and $P\left(Z<1.84\right)=0.9671$ . Substituting these values in the above equation, the required probability is obtained as:

$\begin{array}{c}P\left(1.54<Z<1.84\right)=0.9671-0.9417\\ =0.0254\end{array}$

Thus, the probability is 0.0254.

To determine

The probability that $Z$ is less than 1.57 or greater than 1.84.

Answer to Problem 1PS

The required probability is 0.9746.

Explanation of Solution

The probability that $Z$ is less than 1.57 or greater than 1.84 can be written as;

$P\left(Z<1.57\text{ or }Z>1.84\right)=P\left(Z<1.57\right)+P\left(Z>1.84\right)$

From part (a) and (b), it is obtained that $P\left(Z<1.57\right)=0.9417$ and $P\left(Z>1.84\right)=0.0329$ . Substituting these values in the above equation, the required probability is obtained as:

$\begin{array}{c}P\left(Z<1.57\text{ or }Z>1.84\right)=0.9417+0.0329\\ =0.9746\end{array}$

Thus, the probability is 0.9746.