Chapter 6, Problem 19CQ

Using the standard normal distribution, find each probability.

a. P(0 < z < 2.16)

b. P(−1.87 < z < 0)

c. P(−1.63 < z < 2.17)

d. P(1.72 < z < 1.98)

e. P(−2.17 < z < 0.71)

f. P(z > 1.77)

g. P(z < −2.37)

h. P(z > −1.73)

i. P(z < 2.03)

j. P(z > −1.02)

To determine

To find: The area under the standard normal distribution curve between 0 and 2.16.

That is, $P\left(0<z<2.16\right).$

Answer to Problem 19CQ

The area under the standard normal distribution curve between 0 and 2.16 is 0.4846.

Explanation of Solution

Given info:

The random variable z denote the standard normal variable with parameters $\mu =0,\text{ and }\sigma =1.$

Calculation:

The z-score is 0 is calculated as follows:

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(0\right)$”
• Press Enter.

It returns the probability value is 0.5.

For $P\left(z<2.16\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(2.16\right)$”
• Press Enter.

It returns the probability value is 0.9846.

Solving the equation gives the required probability,

$\begin{array}{c}P\left(0<z<2.16\right)=P\left(z<2.16\right)-P\left(z<0\right)\\ =0.9846-0.5000\\ =0.4846\end{array}$

Conclusion:

The area under the standard normal distribution curve between 0 and 2.16 is 0.4846.

To determine

To find: The area under the standard normal distribution curve between −1.87and 0.

That is, $P\left(-1.87<z<0\right).$

Answer to Problem 19CQ

The area under the standard normal distribution curve between −1.87and 0 is 0.4693.

Explanation of Solution

Calculation:

The z-score is 0 is calculated as follows:

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(0\right)$”
• Press Enter.

It returns the probability value is 0.5.

For $P\left(z<-1.87\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(-1.87\right)$”
• Press Enter.

It returns the probability value is 0.0307.

Solving the equation gives the required probability,

$\begin{array}{c}P\left(-1.87<z<0\right)=P\left(z<0\right)-P\left(z<-1.87\right)\\ =0.5000-0.0307\\ =0.4693\end{array}$

Conclusion:

The area under the standard normal distribution curve between -1.87and 0 is, 0.4693.

To determine

To find: The area under the standard normal distribution curve between -1.63and 2.17.

That is, $P\left(-1.63<z<2.17\right).$

Answer to Problem 19CQ

The area under the standard normal distribution curve between −1.63and 2.17 is 0.9334.

Explanation of Solution

Calculation:

For $P\left(z<-1.63\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(-1.63\right)$”
• Press Enter.

It returns the probability value is 0.0516.

For $P\left(z<2.17\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(2.17\right)$”
• Press Enter.

It returns the probability value is 0.9850.

Solving the equation gives the required probability,

$\begin{array}{c}P\left(-1.63<z<2.17\right)=P\left(z<2.17\right)-P\left(z<-1.63\right)\\ =0.9850-0.0516\\ =0.9334\end{array}$

Conclusion:

Thus, the area under the standard normal distribution curve between −1.63and 2.17 is 0.9334.

To determine

To find: The area under the standard normal distribution curve between 1.72 and 1.98.

That is, $P\left(1.72<z<1.98\right).$

Answer to Problem 19CQ

The area under the standard normal distribution curve between 1.72 and 1.98 is 0.0188.

Explanation of Solution

Calculation:

For $P\left(z<1.72\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(1.72\right)$”
• Press Enter.

It returns the probability value is 0.9573.

For $P\left(z<1.98\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(1.98\right)$”
• Press Enter.

It returns the probability value is 0.9761.

Solving the equation gives the required probability,

$\begin{array}{c}P\left(1.72<z<1.98\right)=P\left(z<1.98\right)-P\left(z<1.72\right)\\ =0.9761-0.9573\\ =0.0188\end{array}$

Conclusion:

Thus, the area under the standard normal distribution curve between 1.72 and 1.98 is 0.0188.

To determine

To find: The area under the standard normal distribution curve between −2.17and 0.71.

That is, $P\left(-2.17<z<0.71\right).$

Answer to Problem 19CQ

The area under the standard normal distribution curve between −2.17and 0.71 is 0.7461.

Explanation of Solution

Given info:

The random variable z denote the standard normal variable with parameters $\mu =0,\text{ and }\sigma =1.$

Calculation:

For $P\left(z<-2.17\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(-2.17\right)$”
• Press Enter.

It returns the probability value is 0.01500.

For $P\left(z<0.71\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(0.71\right)$”
• Press Enter.

It returns the probability value is 0.7611.

Solving the equation gives the required probability,

$\begin{array}{c}P\left(-2.17<z<0.71\right)=P\left(z<0.71\right)-P\left(z<-2.17\right)\\ =0.7611-0.0150\\ =0.7461\end{array}$

Conclusion:

Thus, the area under the standard normal distribution curve between −2.17and 0.71 is 0.7461.

To determine

To find: The area under the normal curve to the right of $z>1.77$.

Answer to Problem 19CQ

The area under the normal curve to the right of $z>1.77$ is 0.0384.

Explanation of Solution

Given info:

The random variable z denote the standard normal variable with parameters $\mu =0,\text{ and }\sigma =1.$

Calculation:

For $P\left(z<1.77\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(1.77\right)$”
• Press Enter.

It returns the probability value is 0.9616.

Solving the equation gives the required probability,

$\begin{array}{c}P\left(z>1.77\right)=1-P\left(z\le 1.77\right)\\ =1-0.9616\\ =0.0384\end{array}$

Conclusion:

Thus, the area under the normal curve to the right of $z>1.77$ is 0.0384.

To determine

To find: The area under the normal curve of $P\left(z<-2.37\right)$

Answer to Problem 19CQ

The area under the normal curve of $P\left(z<-2.37\right)$, 0.0089.

Explanation of Solution

Given info:

The random variable z denote the standard normal variable with parameters $\mu =0,\text{ and }\sigma =1.$

Calculation:

For $P\left(z<-2.37\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(-2.37\right)$”
• Press Enter.

It returns the probability value is 0.0089.

Solving the equation gives the required probability,

$\begin{array}{c}P\left(z<-2.37\right)=P\left(z<-2.03\right)\\ =0.0089\end{array}$

Conclusion:

The area under the normal curve of $P\left(z<-2.37\right)$ is 0.0089.

To determine

To find: The area under the normal curve for $P\left(z>-1.73\right)$.

Answer to Problem 19CQ

The area under the normal curve for $P\left(z>-1.73\right)$ is 0.9582.

Explanation of Solution

Given info:

The random variable z denote the standard normal variable with parameters $\mu =0,\text{ and }\sigma =1.$

Calculation:

For $P\left(z>-1.73\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(-1.73\right)$”
• Press Enter.

It returns the probability value is 0.0418.

$\begin{array}{c}P\left(z>-1.73\right)=1-P\left(z\le -1.73\right)\\ =1-0.0418\\ =0.9582\end{array}$

Conclusion:

Thus, the area under the normal curve for $P\left(z>-1.73\right)$ is 0.9582.

To determine

To find: The area under the normal curve for $P\left(z<-2.03\right)$.

Answer to Problem 19CQ

The area under the normal curve for $P\left(z<2.03\right)$ is 0.9788.

Explanation of Solution

Given info:

The random variable z denote the standard normal variable with parameters $\mu =0,\text{ and }\sigma =1.$

Calculation:

For $P\left(z<2.03\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(2.03\right)$”
• Press Enter.

It returns the probability value is 0.9788

$\begin{array}{c}P\left(z<2.03\right)=P\left(z<2.03\right)\\ =0.9788\end{array}$

Conclusion:

Thus, the area under the normal curve for $P\left(z<2.03\right)$ is 0.9788.

To determine

To find: The area under the normal curve for $P\left(z>-1.02\right)$.

Answer to Problem 19CQ

The area under the normal curve for $P\left(z>-1.02\right)$ is 0.8461.

Explanation of Solution

Given info:

The random variable z denote the standard normal variable with parameters $\mu =0\text{ and }\sigma =1.$

Calculation:

For $P\left(z>-1.02\right)$

Excel procedure:

Step by step procedure to obtain z score using the Excel:

• Select a cell in the Excel
• Type “ $=NORMSDIST\left(-1.02\right)$”
• Press Enter.

It returns the probability value is 0.1539

$\begin{array}{c}P\left(z>-1.02\right)=1-P\left(z\le -1.02\right)\\ =1-0.1539\\ =0.8461\end{array}$

Conclusion:

The area under the normal curve for $P\left(z>-1.02\right)$ is 0.8461.