The Basic Practice of Statistics
The Basic Practice of Statistics
7th Edition
ISBN: 9781464142536
Author: David S. Moore, William I. Notz, Michael A. Fligner
Publisher: W. H. Freeman
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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
Check Mark

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:

The Basic Practice of Statistics, 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)=10.3176=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
Check Mark

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:

The Basic Practice of Statistics, 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]=10.0456=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:

The Basic Practice of Statistics, 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]=10.0026=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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