Chapter 6, Problem 6.1.1RE

Find the area under the standard normal distribution curve for each.

a. Between z = 0 and z = 2.06

b. Between z = 0 and z = 0.53

c. Between z = 1.26 and 1.74

d. Between z = −1.02 and z = 1.63

e. Between z = −0.07 and z = 0.49

To determine

To find: The area under the standard normal distribution curve for $z$ lies between 0 and 2.06. That is, $P\left(0<z<2.06\right).$

Answer to Problem 6.1.1RE

The area under the standard normal distribution curve for $z$ lies between 0 and 2.06 is 0.4803.

Explanation of Solution

Given info:

The z values are 0 and 2.06.

Calculation:

Software procedure:

Step by step procedure to find the area under the normal curve between 0 and 2.06with the help of following instructions:

• Select Graph > Probability Distribution Plot > View Probability >OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0.0 and Standard deviation as 1.0.
• Choose the tab for Shaded Area, then X Value.
• Click the picture for Middle.
• Type in the smaller value 0 for X value1 and then the larger value 2.06 for the X value2.
• Click OK.

Output using the MINITAB software is given below:

Therefore, $P\left(0<z<2.06\right)=\mathrm{0.4803.}$

Conclusion:

The area under the standard normal distribution curve for $z$ lies between 0 and 2.06 is 0.4803

To determine

To find: The area under the standard normal distribution curve for $z$ lies between 0 and 0.53. That is, $P\left(0<z<0.53\right).$

Answer to Problem 6.1.1RE

The area under the standard normal distribution curve for $z$ lies between 0 and 0.53 is 0.2019.

Explanation of Solution

Given info:

The z values are 0 and 0.53.

Calculation:

Software procedure:

Step by step procedure to find the area under the normal curve between 0 and 0.53 with the help of following instructions:

• Select Graph > Probability Distribution Plot > View Probability >OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0.0 and Standard deviation as 1.0.
• Choose the tab for Shaded Area, then X Value.
• Click the picture for Middle.
• Type in the smaller value 0 for X value1 and then the larger value 0.53 for the X value2.
• Click OK.

Output using the MINITAB software is given below:

Therefore, $P\left(0<z<0.53\right)=\mathrm{0.2019.}$

Conclusion:

The area under the standard normal distribution curve for $z$ lies between 0 and 0.53 is 0.2019.

To determine

To find: The area under the standard normal distribution curve for $z$ lies between 1.26 and 1.74. That is, $P\left(1.26<z<1.74\right).$

Answer to Problem 6.1.1RE

The area under the standard normal distribution curve for $z$ lies between 1.26 and 1.74 is 0.0629.

Explanation of Solution

Given info:

The z values are 1.26 and 1.74.

Calculation:

Software procedure:

Step by step procedure to find the area under the normal curve between 1.26 and 1.74 with the help of following instructions:

• Select Graph > Probability Distribution Plot > View Probability >OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0.0 and Standard deviation as 1.0.
• Choose the tab for Shaded Area, then X Value.
• Click the picture for Middle.
• Type in the smaller value 1.26 for X value1 and then the larger value 1.74 for the X value2.
• Click OK.

Output using the MINITAB software is given below:

Therefore, $P\left(1.26<z<1.74\right)=\mathrm{0.0629.}$

Conclusion:

The area under the standard normal distribution curve for $z$ lies between 1.26 and 1.74 is 0.0629.

To determine

To find: The area under the standard normal distribution curve for $z$ lies between –1.02 and 1.63. That is, $P\left(-1.02<z<1.63\right).$

Answer to Problem 6.1.1RE

The area under the standard normal distribution curve for $z$ lies between –1.02 and 1.63 is 0.7945.

Explanation of Solution

Given info:

The z values are –1.02 and 1.63.

Calculation:

Software procedure:

Step by step procedure to; find the area under the normal curve between -1.02 and 1.63 with the help of following instructions:

• Select Graph > Probability Distribution Plot > View Probability >OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0.0 and Standard deviation as 1.0.
• Choose the tab for Shaded Area, then X Value.
• Click the picture for Middle.
• Type in the smaller value −1.02 for X value1 and then the larger value 1.63 for the X value2.
• Click OK.

Output using the MINITAB software is given below:

Therefore, $P\left(-1.02<z<1.63\right)=0.7945$

Conclusion:

The area under the standard normal distribution curve for $z$ lies between −1.02 and 1.63 is 0.7945.

To determine

To find: The area under the standard normal distribution curve for $z$ lies between -0.07 and 0.49. That is, $P\left(-0.07<z<0.49\right).$

Answer to Problem 6.1.1RE

The area under the standard normal distribution curve for $z$ lies between −0.07 and 0.49 is 0.2158.

Explanation of Solution

Given info:

The z values are -0.07 and 0.49.

Calculation:

Software procedure:

Step by step procedure to; find the area under the normal curve between -0.07 and 0.49 with the help of following instructions:

• Select Graph > Probability Distribution Plot > View Probability >OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0.0 and Standard deviation as 1.0.
• Choose the tab for Shaded Area, then X Value.
• Click the picture for Middle.
• Type in the smaller value −0.07 for X value1 and then the larger value 0.49 for the X value2.
• Click OK.

Output using the MINITAB software is given below:

Therefore, $P\left(-0.07<z<0.49\right)=\mathrm{0.2158.}$

Conclusion:

The area under the standard normal distribution curve for $z$ lies between −0.07 and 0.49 is 0.2158.