Beginning Statistics, 2nd Edition
Beginning Statistics, 2nd Edition
2nd Edition
ISBN: 9781932628678
Author: Carolyn Warren; Kimberly Denley; Emily Atchley
Publisher: Hawkes Learning Systems
Question
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Chapter 6.2, Problem 21E
To determine

To find:

The difference between the more precise value found using the normal distribution tables or technology and the approximation given by the Empirical Rule for the area under the standard normal curve within two standard deviations of the mean.

Also the similar difference for the area within three standard deviations of the mean.

Expert Solution & Answer
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Answer to Problem 21E

Solution:

The difference of area under the standard normal curve between the precise calculation and Empirical Rule within two standard deviations is 0.45% and within three standard deviations is 0.03% of the total area under the curve.

Explanation of Solution

Given:

The ranges of values are (μ2σ,μ+2σ) and (μ3σ,μ+3σ).

Description:

In statistics, the normal distribution probability density function is given by:

f(x|μ,σ2)=12πσ2e(xμ)22σ2

where, x denotes random variable value while μ is the mean and the variance is σ2.

The normal Gaussian distribution with the following characteristics:

μ=0, and

σ=1

is called standard normal distribution with probability density function:

f(x|0,1)=12πex22

When this function is plotted on the x-axis, the area under between the curve and x-axis is equal to one, because the sum of probabilities is unity.

Area on the left of given z-value (say z=za) is the probability P(Xa) given by the integration:

af(x)dx.

The z-score is obtained as the number of unit standard deviations the x-value is above or is below the mean.

The area beyond two z-values, the smaller being z1 and larger being z2 is:

P(z1zz2)=P(zz2)P(zz1)

The Empirical Rule, also known as the 689599.7 rule states that the percentage of area between standard normal curve and x-axis is 68% between z=1 and z=1, 95% between z=2 and z=2, and 99.7% between z=3 and z=3.

Calculation:

First, consider z1 as the value at two standard deviations less than mean and z2 as two standard deviations above mean.

Next, consider z1 as the value at three standard deviations less than mean and z2 as three standard deviations above mean.

The required probabilities are the differences of the probabilities to the left of z1 and probabilities on the left of z2.

The z-value corresponding to μ2σ is z1=2 and that corresponding to μ+2σ is z2=2.

The area under standard normal curve beyond the z-values is calculated as:

P(2z2)=P(z2)P(z2)=0.977250.02275=0.9545

This is the precise probability as found using standard normal table.

The Empirical Rule states that between z=2 and z=2 the area, and hence probability is 0.95.

The difference is as:

0.95450.95=0.0045

which can be changed into the percent as:

(0.0045×100)%=0.45% of the area.

On the other hand, the z-value corresponding to μ3σ is z1=3 and that corresponding to μ+3σ is z2=3.

The area under standard normal curve beyond the z-values is calculated as:

P(3z3)=P(z3)P(z3)=0.998650.00135=0.9973

This is the precise probability as found using standard normal table.

The Empirical Rule states that between z=3 and z=3 the area, and hence probability is 0.997.

The difference is as:

0.99730.997=0.0003

which can be changed into the percent as:

(0.0003×100)%=0.03% of the area.

Conclusion:

The difference of area under the standard normal curve between the precise calculation and Empirical Rule within two standard deviations is 0.45% and within three standard deviations is 0.03% of the total area under the curve.

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Chapter 6 Solutions

Beginning Statistics, 2nd Edition

Ch. 6.1 - Prob. 11ECh. 6.1 - Prob. 12ECh. 6.1 - Prob. 13ECh. 6.1 - Prob. 14ECh. 6.1 - Prob. 15ECh. 6.1 - Prob. 16ECh. 6.1 - Prob. 17ECh. 6.1 - Prob. 18ECh. 6.1 - Prob. 19ECh. 6.1 - Prob. 20ECh. 6.1 - Prob. 21ECh. 6.1 - Prob. 22ECh. 6.1 - Prob. 23ECh. 6.1 - Prob. 24ECh. 6.1 - Prob. 25ECh. 6.2 - Prob. 1ECh. 6.2 - Prob. 2ECh. 6.2 - Prob. 3ECh. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.2 - Prob. 7ECh. 6.2 - Prob. 8ECh. 6.2 - Prob. 9ECh. 6.2 - Prob. 10ECh. 6.2 - Prob. 11ECh. 6.2 - Prob. 12ECh. 6.2 - Prob. 13ECh. 6.2 - Prob. 14ECh. 6.2 - Prob. 15ECh. 6.2 - Prob. 16ECh. 6.2 - Prob. 17ECh. 6.2 - Prob. 18ECh. 6.2 - Prob. 19ECh. 6.2 - Prob. 20ECh. 6.2 - Prob. 21ECh. 6.2 - Prob. 22ECh. 6.2 - Prob. 23ECh. 6.2 - Prob. 24ECh. 6.2 - Prob. 25ECh. 6.2 - Prob. 26ECh. 6.2 - Prob. 27ECh. 6.2 - Prob. 28ECh. 6.2 - Prob. 29ECh. 6.2 - Prob. 30ECh. 6.2 - Prob. 31ECh. 6.2 - Prob. 32ECh. 6.2 - Prob. 33ECh. 6.2 - Prob. 34ECh. 6.2 - Prob. 35ECh. 6.2 - Prob. 36ECh. 6.2 - Prob. 37ECh. 6.2 - Prob. 38ECh. 6.2 - Prob. 39ECh. 6.2 - Prob. 40ECh. 6.2 - Prob. 41ECh. 6.2 - Prob. 42ECh. 6.2 - Prob. 43ECh. 6.2 - Prob. 44ECh. 6.2 - Prob. 45ECh. 6.2 - Prob. 46ECh. 6.2 - Prob. 47ECh. 6.2 - Prob. 48ECh. 6.2 - Prob. 49ECh. 6.3 - Prob. 1ECh. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.3 - Prob. 4ECh. 6.3 - Prob. 5ECh. 6.3 - Prob. 6ECh. 6.3 - Prob. 7ECh. 6.3 - Prob. 8ECh. 6.3 - Prob. 9ECh. 6.3 - Prob. 10ECh. 6.3 - Prob. 11ECh. 6.4 - Prob. 1ECh. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Prob. 9ECh. 6.4 - Prob. 10ECh. 6.4 - Prob. 11ECh. 6.4 - Prob. 12ECh. 6.4 - Prob. 13ECh. 6.4 - Prob. 14ECh. 6.4 - Prob. 15ECh. 6.4 - Prob. 16ECh. 6.4 - Prob. 17ECh. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - Prob. 20ECh. 6.4 - Prob. 21ECh. 6.4 - Prob. 22ECh. 6.4 - Prob. 23ECh. 6.4 - Prob. 24ECh. 6.4 - Prob. 25ECh. 6.4 - Prob. 26ECh. 6.4 - Prob. 27ECh. 6.4 - Prob. 28ECh. 6.4 - Prob. 29ECh. 6.5 - Prob. 1ECh. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Prob. 8ECh. 6.5 - Prob. 9ECh. 6.5 - Prob. 10ECh. 6.5 - Prob. 11ECh. 6.5 - Prob. 12ECh. 6.5 - Prob. 13ECh. 6.5 - Prob. 14ECh. 6.5 - Prob. 15ECh. 6.5 - Prob. 16ECh. 6.5 - Prob. 17ECh. 6.5 - Prob. 18ECh. 6.5 - Prob. 19ECh. 6.5 - Prob. 20ECh. 6.5 - Prob. 21ECh. 6.5 - Prob. 22ECh. 6.5 - Prob. 23ECh. 6.5 - Prob. 24ECh. 6.5 - Prob. 25ECh. 6.5 - Prob. 26ECh. 6.5 - Prob. 27ECh. 6.5 - Prob. 28ECh. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.CR - Prob. 1CRCh. 6.CR - Prob. 2CRCh. 6.CR - Prob. 3CRCh. 6.CR - Prob. 4CRCh. 6.CR - Prob. 5CRCh. 6.CR - Prob. 6CRCh. 6.CR - Prob. 7CRCh. 6.CR - Prob. 8CRCh. 6.CR - Prob. 9CRCh. 6.CR - Prob. 10CRCh. 6.CR - Prob. 11CRCh. 6.CR - Prob. 12CRCh. 6.CR - Prob. 13CRCh. 6.CR - Prob. 14CRCh. 6.CR - Prob. 15CRCh. 6.CR - Prob. 16CRCh. 6.CR - Prob. 17CRCh. 6.CR - Prob. 18CRCh. 6.CR - Prob. 19CRCh. 6.CR - Prob. 20CRCh. 6.CR - Prob. 21CRCh. 6.CR - Prob. 22CRCh. 6.P - Prob. 1PCh. 6.P - Prob. 2PCh. 6.P - Prob. 3PCh. 6.P - Prob. 4PCh. 6.P - Prob. 5PCh. 6.P - Prob. 6PCh. 6.P - Prob. 7PCh. 6.P - Prob. 8P
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