The area within 1 standard deviation of the mean. To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule .

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Answer 1

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 3, Problem 3.52E

a.

To determine

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

a.

Expert Solution

Answer to Problem 3.52E

The area within 1 standard deviation of the mean is 0.6826.

The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Given info:

The mean and standard deviation of the normal distribution are 0 and 1.

68-95-99.7 Rule:

About 68% of the observations fall within σ of the mean μ

About 95% of the observations fall within 2σ of the mean μ

About 99.7% of the observations fall within 3σ of the mean μ

Calculation:

Find the area of 1 standard deviation on either side of the mean using the normal density curve applet.

Applet procedure:

Step-by-step Applet procedure to find the area of 1 standard deviation on either side of the mean is given as follows:

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 1 standard deviation on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 1

From the Applet output, the area of one standard deviation below −1 is 0.1587 and the above 1 is 0.1587.

The formula to find the area within 1 standard deviation of the mean is,

[Area within 1 standard deviations of the mean] = 1−(Area to the left of (−1)+Area to the right of 1)=1−(0.1587+0.1587)=1−0.3176=0.6826

Thus, the area within 1 standard deviation of the mean is 0.6826. That is, about 68% of the observations fall within σ of the mean μ .

From the 68-95-99.7 rule, the area within 1 standard deviation of the mean is about 68%.

Justification:

The result of the area within 1 standard deviation of the mean is approximately same by using the applet and the 68-95-99.7 rule.

b.

To determine

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

b.

Expert Solution

Answer to Problem 3.52E

The area within 2 standard deviations of the mean is 0.9544 and the area within 3 standard deviations of the mean is 0.9974.

The area within 2 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Calculation:

Finding the area of 2 standard deviations on either side of the mean:

Applet procedure:

Step-by-step Applet procedure to find the area of 2 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 2 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 2

From the Applet output, the area to the left of −2 is 0.0228 and the area to right of 2 is 0.0228.

The formula to find the area within 2 standard deviations of the mean is,

[Area within 2 standard deviations of the mean ]= 1−[Area to the left of (−2)+Area to the right of 2]=1−[0.0228+0.0228]=1−0.0456=0.9544

Thus, the area within 2 standard deviations of the mean is 0.9544. That is, about 95% of the observations fall within 2σ of the mean μ .

From the 68-95-99.7 rule, the area within 2 standard deviations of the mean is about 95%.

Finding the area of 3 standard deviations on either side of the mean:

Applet Procedure:

Step-by-step Applet procedure to find the area of 3 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 3 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 3

From the Applet output, the area to the left of −3 is 0.0013 and the area to right of 3 is 0.0013.

The formula to find the area within 3 standard deviations of the mean is,

[Area within 3 standard deviations of the mean ]= 1−[Area to the left of (−3)+Area to the right of 3]=1−[0.0013+0.0013]=1−0.0026=0.9974

Thus, the area within 3 standard deviations of the mean is 0.9974. That is, about 99.7% of the observations fall within 3σ of the mean μ .

From the 68-95-99.7 rule, the area within 3 standard deviations of the mean is about 99.7%.

Justification:

The result of the area within 2 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule and also the result of the area within 3 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule.

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Answer 2

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 3, Problem 3.52E

a.

To determine

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

a.

Expert Solution

Answer to Problem 3.52E

The area within 1 standard deviation of the mean is 0.6826.

The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Given info:

The mean and standard deviation of the normal distribution are 0 and 1.

68-95-99.7 Rule:

About 68% of the observations fall within σ of the mean μ

About 95% of the observations fall within 2σ of the mean μ

About 99.7% of the observations fall within 3σ of the mean μ

Calculation:

Find the area of 1 standard deviation on either side of the mean using the normal density curve applet.

Applet procedure:

Step-by-step Applet procedure to find the area of 1 standard deviation on either side of the mean is given as follows:

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 1 standard deviation on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 1

From the Applet output, the area of one standard deviation below −1 is 0.1587 and the above 1 is 0.1587.

The formula to find the area within 1 standard deviation of the mean is,

[Area within 1 standard deviations of the mean] = 1−(Area to the left of (−1)+Area to the right of 1)=1−(0.1587+0.1587)=1−0.3176=0.6826

Thus, the area within 1 standard deviation of the mean is 0.6826. That is, about 68% of the observations fall within σ of the mean μ .

From the 68-95-99.7 rule, the area within 1 standard deviation of the mean is about 68%.

Justification:

The result of the area within 1 standard deviation of the mean is approximately same by using the applet and the 68-95-99.7 rule.

b.

To determine

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

b.

Expert Solution

Answer to Problem 3.52E

The area within 2 standard deviations of the mean is 0.9544 and the area within 3 standard deviations of the mean is 0.9974.

The area within 2 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Calculation:

Finding the area of 2 standard deviations on either side of the mean:

Applet procedure:

Step-by-step Applet procedure to find the area of 2 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 2 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 2

From the Applet output, the area to the left of −2 is 0.0228 and the area to right of 2 is 0.0228.

The formula to find the area within 2 standard deviations of the mean is,

[Area within 2 standard deviations of the mean ]= 1−[Area to the left of (−2)+Area to the right of 2]=1−[0.0228+0.0228]=1−0.0456=0.9544

Thus, the area within 2 standard deviations of the mean is 0.9544. That is, about 95% of the observations fall within 2σ of the mean μ .

From the 68-95-99.7 rule, the area within 2 standard deviations of the mean is about 95%.

Finding the area of 3 standard deviations on either side of the mean:

Applet Procedure:

Step-by-step Applet procedure to find the area of 3 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 3 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 3

From the Applet output, the area to the left of −3 is 0.0013 and the area to right of 3 is 0.0013.

The formula to find the area within 3 standard deviations of the mean is,

[Area within 3 standard deviations of the mean ]= 1−[Area to the left of (−3)+Area to the right of 3]=1−[0.0013+0.0013]=1−0.0026=0.9974

Thus, the area within 3 standard deviations of the mean is 0.9974. That is, about 99.7% of the observations fall within 3σ of the mean μ .

From the 68-95-99.7 rule, the area within 3 standard deviations of the mean is about 99.7%.

Justification:

The result of the area within 2 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule and also the result of the area within 3 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule.

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Answer 3

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 3, Problem 3.52E

a.

To determine

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

a.

Expert Solution

Answer to Problem 3.52E

The area within 1 standard deviation of the mean is 0.6826.

The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Given info:

The mean and standard deviation of the normal distribution are 0 and 1.

68-95-99.7 Rule:

About 68% of the observations fall within σ of the mean μ

About 95% of the observations fall within 2σ of the mean μ

About 99.7% of the observations fall within 3σ of the mean μ

Calculation:

Find the area of 1 standard deviation on either side of the mean using the normal density curve applet.

Applet procedure:

Step-by-step Applet procedure to find the area of 1 standard deviation on either side of the mean is given as follows:

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 1 standard deviation on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 1

From the Applet output, the area of one standard deviation below −1 is 0.1587 and the above 1 is 0.1587.

The formula to find the area within 1 standard deviation of the mean is,

[Area within 1 standard deviations of the mean] = 1−(Area to the left of (−1)+Area to the right of 1)=1−(0.1587+0.1587)=1−0.3176=0.6826

Thus, the area within 1 standard deviation of the mean is 0.6826. That is, about 68% of the observations fall within σ of the mean μ .

From the 68-95-99.7 rule, the area within 1 standard deviation of the mean is about 68%.

Justification:

The result of the area within 1 standard deviation of the mean is approximately same by using the applet and the 68-95-99.7 rule.

b.

To determine

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

b.

Expert Solution

Answer to Problem 3.52E

The area within 2 standard deviations of the mean is 0.9544 and the area within 3 standard deviations of the mean is 0.9974.

The area within 2 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Calculation:

Finding the area of 2 standard deviations on either side of the mean:

Applet procedure:

Step-by-step Applet procedure to find the area of 2 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 2 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 2

From the Applet output, the area to the left of −2 is 0.0228 and the area to right of 2 is 0.0228.

The formula to find the area within 2 standard deviations of the mean is,

[Area within 2 standard deviations of the mean ]= 1−[Area to the left of (−2)+Area to the right of 2]=1−[0.0228+0.0228]=1−0.0456=0.9544

Thus, the area within 2 standard deviations of the mean is 0.9544. That is, about 95% of the observations fall within 2σ of the mean μ .

From the 68-95-99.7 rule, the area within 2 standard deviations of the mean is about 95%.

Finding the area of 3 standard deviations on either side of the mean:

Applet Procedure:

Step-by-step Applet procedure to find the area of 3 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 3 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 3

From the Applet output, the area to the left of −3 is 0.0013 and the area to right of 3 is 0.0013.

The formula to find the area within 3 standard deviations of the mean is,

[Area within 3 standard deviations of the mean ]= 1−[Area to the left of (−3)+Area to the right of 3]=1−[0.0013+0.0013]=1−0.0026=0.9974

Thus, the area within 3 standard deviations of the mean is 0.9974. That is, about 99.7% of the observations fall within 3σ of the mean μ .

From the 68-95-99.7 rule, the area within 3 standard deviations of the mean is about 99.7%.

Justification:

The result of the area within 2 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule and also the result of the area within 3 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule.

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Answer 4

Question

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 3, Problem 3.52E

a.

To determine

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

a.

Expert Solution

Answer to Problem 3.52E

The area within 1 standard deviation of the mean is 0.6826.

The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Given info:

The mean and standard deviation of the normal distribution are 0 and 1.

68-95-99.7 Rule:

About 68% of the observations fall within σ of the mean μ

About 95% of the observations fall within 2σ of the mean μ

About 99.7% of the observations fall within 3σ of the mean μ

Calculation:

Find the area of 1 standard deviation on either side of the mean using the normal density curve applet.

Applet procedure:

Step-by-step Applet procedure to find the area of 1 standard deviation on either side of the mean is given as follows:

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 1 standard deviation on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 1

From the Applet output, the area of one standard deviation below −1 is 0.1587 and the above 1 is 0.1587.

The formula to find the area within 1 standard deviation of the mean is,

[Area within 1 standard deviations of the mean] = 1−(Area to the left of (−1)+Area to the right of 1)=1−(0.1587+0.1587)=1−0.3176=0.6826

Thus, the area within 1 standard deviation of the mean is 0.6826. That is, about 68% of the observations fall within σ of the mean μ .

From the 68-95-99.7 rule, the area within 1 standard deviation of the mean is about 68%.

Justification:

The result of the area within 1 standard deviation of the mean is approximately same by using the applet and the 68-95-99.7 rule.

b.

To determine

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

b.

Expert Solution

Answer to Problem 3.52E

The area within 2 standard deviations of the mean is 0.9544 and the area within 3 standard deviations of the mean is 0.9974.

The area within 2 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Calculation:

Finding the area of 2 standard deviations on either side of the mean:

Applet procedure:

Step-by-step Applet procedure to find the area of 2 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 2 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 2

From the Applet output, the area to the left of −2 is 0.0228 and the area to right of 2 is 0.0228.

The formula to find the area within 2 standard deviations of the mean is,

[Area within 2 standard deviations of the mean ]= 1−[Area to the left of (−2)+Area to the right of 2]=1−[0.0228+0.0228]=1−0.0456=0.9544

Thus, the area within 2 standard deviations of the mean is 0.9544. That is, about 95% of the observations fall within 2σ of the mean μ .

From the 68-95-99.7 rule, the area within 2 standard deviations of the mean is about 95%.

Finding the area of 3 standard deviations on either side of the mean:

Applet Procedure:

Step-by-step Applet procedure to find the area of 3 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 3 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 3

From the Applet output, the area to the left of −3 is 0.0013 and the area to right of 3 is 0.0013.

The formula to find the area within 3 standard deviations of the mean is,

[Area within 3 standard deviations of the mean ]= 1−[Area to the left of (−3)+Area to the right of 3]=1−[0.0013+0.0013]=1−0.0026=0.9974

Thus, the area within 3 standard deviations of the mean is 0.9974. That is, about 99.7% of the observations fall within 3σ of the mean μ .

From the 68-95-99.7 rule, the area within 3 standard deviations of the mean is about 99.7%.

Justification:

The result of the area within 2 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule and also the result of the area within 3 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule.

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Answer 5

Chapter 3, Problem 3.52E

a.

To determine

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

a.

Expert Solution

Answer to Problem 3.52E

The area within 1 standard deviation of the mean is 0.6826.

The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Given info:

The mean and standard deviation of the normal distribution are 0 and 1.

68-95-99.7 Rule:

About 68% of the observations fall within σ of the mean μ

About 95% of the observations fall within 2σ of the mean μ

About 99.7% of the observations fall within 3σ of the mean μ

Calculation:

Find the area of 1 standard deviation on either side of the mean using the normal density curve applet.

Applet procedure:

Step-by-step Applet procedure to find the area of 1 standard deviation on either side of the mean is given as follows:

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 1 standard deviation on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 1

From the Applet output, the area of one standard deviation below −1 is 0.1587 and the above 1 is 0.1587.

The formula to find the area within 1 standard deviation of the mean is,

[Area within 1 standard deviations of the mean] = 1−(Area to the left of (−1)+Area to the right of 1)=1−(0.1587+0.1587)=1−0.3176=0.6826

Thus, the area within 1 standard deviation of the mean is 0.6826. That is, about 68% of the observations fall within σ of the mean μ .

From the 68-95-99.7 rule, the area within 1 standard deviation of the mean is about 68%.

Justification:

The result of the area within 1 standard deviation of the mean is approximately same by using the applet and the 68-95-99.7 rule.

b.

To determine

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

b.

Expert Solution

Answer to Problem 3.52E

The area within 2 standard deviations of the mean is 0.9544 and the area within 3 standard deviations of the mean is 0.9974.

The area within 2 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Calculation:

Finding the area of 2 standard deviations on either side of the mean:

Applet procedure:

Step-by-step Applet procedure to find the area of 2 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 2 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 2

From the Applet output, the area to the left of −2 is 0.0228 and the area to right of 2 is 0.0228.

The formula to find the area within 2 standard deviations of the mean is,

[Area within 2 standard deviations of the mean ]= 1−[Area to the left of (−2)+Area to the right of 2]=1−[0.0228+0.0228]=1−0.0456=0.9544

Thus, the area within 2 standard deviations of the mean is 0.9544. That is, about 95% of the observations fall within 2σ of the mean μ .

From the 68-95-99.7 rule, the area within 2 standard deviations of the mean is about 95%.

Finding the area of 3 standard deviations on either side of the mean:

Applet Procedure:

Step-by-step Applet procedure to find the area of 3 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 3 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 3

From the Applet output, the area to the left of −3 is 0.0013 and the area to right of 3 is 0.0013.

The formula to find the area within 3 standard deviations of the mean is,

[Area within 3 standard deviations of the mean ]= 1−[Area to the left of (−3)+Area to the right of 3]=1−[0.0013+0.0013]=1−0.0026=0.9974

Thus, the area within 3 standard deviations of the mean is 0.9974. That is, about 99.7% of the observations fall within 3σ of the mean μ .

From the 68-95-99.7 rule, the area within 3 standard deviations of the mean is about 99.7%.

Justification:

The result of the area within 2 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule and also the result of the area within 3 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule.

Answer 6

Chapter 3, Problem 3.52E

a.

To determine

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

a.

Expert Solution

Answer to Problem 3.52E

The area within 1 standard deviation of the mean is 0.6826.

The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Given info:

The mean and standard deviation of the normal distribution are 0 and 1.

68-95-99.7 Rule:

About 68% of the observations fall within σ of the mean μ

About 95% of the observations fall within 2σ of the mean μ

About 99.7% of the observations fall within 3σ of the mean μ

Calculation:

Find the area of 1 standard deviation on either side of the mean using the normal density curve applet.

Applet procedure:

Step-by-step Applet procedure to find the area of 1 standard deviation on either side of the mean is given as follows:

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 1 standard deviation on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 1

From the Applet output, the area of one standard deviation below −1 is 0.1587 and the above 1 is 0.1587.

The formula to find the area within 1 standard deviation of the mean is,

[Area within 1 standard deviations of the mean] = 1−(Area to the left of (−1)+Area to the right of 1)=1−(0.1587+0.1587)=1−0.3176=0.6826

Thus, the area within 1 standard deviation of the mean is 0.6826. That is, about 68% of the observations fall within σ of the mean μ .

From the 68-95-99.7 rule, the area within 1 standard deviation of the mean is about 68%.

Justification:

The result of the area within 1 standard deviation of the mean is approximately same by using the applet and the 68-95-99.7 rule.

Answer 7

Chapter 3, Problem 3.52E

a.

To determine

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

Answer 8

a.

Expert Solution

Answer to Problem 3.52E

The area within 1 standard deviation of the mean is 0.6826.

The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Given info:

The mean and standard deviation of the normal distribution are 0 and 1.

68-95-99.7 Rule:

About 68% of the observations fall within σ of the mean μ

About 95% of the observations fall within 2σ of the mean μ

About 99.7% of the observations fall within 3σ of the mean μ

Calculation:

Find the area of 1 standard deviation on either side of the mean using the normal density curve applet.

Applet procedure:

Step-by-step Applet procedure to find the area of 1 standard deviation on either side of the mean is given as follows:

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 1 standard deviation on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 1

From the Applet output, the area of one standard deviation below −1 is 0.1587 and the above 1 is 0.1587.

The formula to find the area within 1 standard deviation of the mean is,

[Area within 1 standard deviations of the mean] = 1−(Area to the left of (−1)+Area to the right of 1)=1−(0.1587+0.1587)=1−0.3176=0.6826

Thus, the area within 1 standard deviation of the mean is 0.6826. That is, about 68% of the observations fall within σ of the mean μ .

From the 68-95-99.7 rule, the area within 1 standard deviation of the mean is about 68%.

Justification:

The result of the area within 1 standard deviation of the mean is approximately same by using the applet and the 68-95-99.7 rule.

Answer 13

b.

To determine

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

b.

Expert Solution

Answer to Problem 3.52E

The area within 2 standard deviations of the mean is 0.9544 and the area within 3 standard deviations of the mean is 0.9974.

The area within 2 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Explanation of Solution

Calculation:

Finding the area of 2 standard deviations on either side of the mean:

Applet procedure:

Step-by-step Applet procedure to find the area of 2 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 2 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 2

From the Applet output, the area to the left of −2 is 0.0228 and the area to right of 2 is 0.0228.

The formula to find the area within 2 standard deviations of the mean is,

[Area within 2 standard deviations of the mean ]= 1−[Area to the left of (−2)+Area to the right of 2]=1−[0.0228+0.0228]=1−0.0456=0.9544

Thus, the area within 2 standard deviations of the mean is 0.9544. That is, about 95% of the observations fall within 2σ of the mean μ .

From the 68-95-99.7 rule, the area within 2 standard deviations of the mean is about 95%.

Finding the area of 3 standard deviations on either side of the mean:

Applet Procedure:

Step-by-step Applet procedure to find the area of 3 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 3 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 3

From the Applet output, the area to the left of −3 is 0.0013 and the area to right of 3 is 0.0013.

The formula to find the area within 3 standard deviations of the mean is,

[Area within 3 standard deviations of the mean ]= 1−[Area to the left of (−3)+Area to the right of 3]=1−[0.0013+0.0013]=1−0.0026=0.9974

Thus, the area within 3 standard deviations of the mean is 0.9974. That is, about 99.7% of the observations fall within 3σ of the mean μ .

From the 68-95-99.7 rule, the area within 3 standard deviations of the mean is about 99.7%.

Justification:

The result of the area within 2 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule and also the result of the area within 3 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule.

Answer 14

b.

To determine

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

Answer 15

b.

Expert Solution

Answer to Problem 3.52E

The area within 2 standard deviations of the mean is 0.9544 and the area within 3 standard deviations of the mean is 0.9974.

The area within 2 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule is both approximately equal.

Answer 16

b.

Expert Solution

Answer 17

b.

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Calculation:

Finding the area of 2 standard deviations on either side of the mean:

Applet procedure:

Step-by-step Applet procedure to find the area of 2 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 2 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 2

From the Applet output, the area to the left of −2 is 0.0228 and the area to right of 2 is 0.0228.

The formula to find the area within 2 standard deviations of the mean is,

[Area within 2 standard deviations of the mean ]= 1−[Area to the left of (−2)+Area to the right of 2]=1−[0.0228+0.0228]=1−0.0456=0.9544

Thus, the area within 2 standard deviations of the mean is 0.9544. That is, about 95% of the observations fall within 2σ of the mean μ .

From the 68-95-99.7 rule, the area within 2 standard deviations of the mean is about 95%.

Finding the area of 3 standard deviations on either side of the mean:

Applet Procedure:

Step-by-step Applet procedure to find the area of 3 standard deviations on either side of the mean is given as follows

In Statistical Applets, choose Normal Density Curve.
In the normal density curve drag one flag and place at 3 standard deviations on either side of the mean.

Output obtained from Applet:

Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access), Chapter 3, Problem 3.52E , additional homework tip 3

From the Applet output, the area to the left of −3 is 0.0013 and the area to right of 3 is 0.0013.

The formula to find the area within 3 standard deviations of the mean is,

[Area within 3 standard deviations of the mean ]= 1−[Area to the left of (−3)+Area to the right of 3]=1−[0.0013+0.0013]=1−0.0026=0.9974

Thus, the area within 3 standard deviations of the mean is 0.9974. That is, about 99.7% of the observations fall within 3σ of the mean μ .

From the 68-95-99.7 rule, the area within 3 standard deviations of the mean is about 99.7%.

Justification:

The result of the area within 2 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule and also the result of the area within 3 standard deviations of the mean is approximately same by using the applet and the 68-95-99.7 rule.

Answer 20

Ch. 3.1 - Prob. 3.1AYK Ch. 3.1 - Prob. 3.2AYK Ch. 3.2 - Prob. 3.3AYK Ch. 3.2 - Prob. 3.4AYK Ch. 3.4 - Prob. 3.5AYK Ch. 3.4 - Prob. 3.6AYK Ch. 3.4 - Prob. 3.7AYK Ch. 3.5 - Prob. 3.8AYK Ch. 3.5 - Prob. 3.9AYK Ch. 3.7 - Prob. 3.10AYK

The area within 1 standard deviation of the mean. To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule .

Concept explainers

Videos

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

Answer to Problem 3.52E

Explanation of Solution

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

Answer to Problem 3.52E

Explanation of Solution

Want to see more full solutions like this?

Chapter 3 Solutions

The area within 1 standard deviation of the mean. To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule .

Concept explainers

Videos

To explain: The area within 1 standard deviation of the mean. To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

Answer to Problem 3.52E

Explanation of Solution

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean. To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.

Answer to Problem 3.52E

Explanation of Solution

Want to see more full solutions like this?

Chapter 3 Solutions

To explain: The area within 1 standard deviation of the mean.

To compare: The area within 1 standard deviation of the mean by using applet and 68-95-99.7 rule.

To explain: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean.

To compare: The area within 2 standard deviations of the mean and the area within 3 standard deviations of the mean by using applet and 68-95-99.7 rule.