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.

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 $\left(\mu -2\sigma ,\mu +2\sigma \right)$ and $\left(\mu -3\sigma ,\mu +3\sigma \right)$.

Description:

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

$f\left(x|\mu ,{\sigma }^2\right)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}}$

where, $x$ denotes random variable value while $\mu$ is the mean and the variance is ${\sigma }^2$.

The normal Gaussian distribution with the following characteristics:

$\mu =0$, and

$\sigma =1$

is called standard normal distribution with probability density function:

$f\left(x|0,1\right)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}$

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 $z=z_a$) is the probability $P\left(X\le a\right)$ given by the integration:

${\displaystyle \underset{-\infty }{\overset{a}{\int }}f\left(x\right)dx}$.

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 $z_1$ and larger being $z_2$ is:

$P\left(z_1\le z\le z_2\right)=P\left(z\le z_2\right)-P\left(z\le z_1\right)$

The Empirical Rule, also known as the $68-95-99.7$ 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 $z_1$ as the value at two standard deviations less than mean and $z_2$ as two standard deviations above mean.

Next, consider $z_1$ as the value at three standard deviations less than mean and $z_2$ as three standard deviations above mean.

The required probabilities are the differences of the probabilities to the left of $z_1$ and probabilities on the left of $z_2$.

The $z$-value corresponding to $\mu -2\sigma$ is $z_1=-2$ and that corresponding to $\mu +2\sigma$ is $z_2=2$.

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

$\begin{array}{rll}P\left(-2\le z\le 2\right)& =& P\left(z\le 2\right)-P\left(z\le -2\right)\\ & =& 0.97725-0.02275\\ & =& 0.9545\end{array}$

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:

$0.9545-0.95=0.0045$

which can be changed into the percent as:

$\left(0.0045\times 100\right)\%=0.45\%$ of the area.

On the other hand, the $z$-value corresponding to $\mu -3\sigma$ is $z_1=-3$ and that corresponding to $\mu +3\sigma$ is $z_2=3$.

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

$\begin{array}{rll}P\left(-3\le z\le 3\right)& =& P\left(z\le 3\right)-P\left(z\le -3\right)\\ & =& 0.99865-0.00135\\ & =& 0.9973\end{array}$

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:

$0.9973-0.997=0.0003$

which can be changed into the percent as:

$\left(0.0003\times 100\right)\%=0.03\%$ 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.