Chapter 8, Problem 42RE

a.

To determine

To find: The proportion of the population lying within one standard deviation of the mean is close to $68\%$ .

Answer to Problem 42RE

The proportion of the population lying within one standard deviation of the mean is close to $60\%$ is $0.6827$ .

Explanation of Solution

Given:

The given information is $\mu =0,\sigma =1$ and integrate from $-1$ to $1$ .

Calculation:

For the proportion of the population lying within $1$ standard deviation,

  $\underset{-1}{\overset{1}{{\displaystyle \int \frac{1}{\sqrt{2\pi }}}}}{e}^{\frac{-x^2}{2}}dx\approx 0.6827$ .

Therefore, the proportion of the population lying within one standard deviation of the mean is close to $60\%$ is $0.6827$ .

b.

To determine

To find: The remaining parts of the rule.

Answer to Problem 42RE

The answer is shown below.

Explanation of Solution

Given:

The given information is $\mu =0,\sigma =1$ and integrate from $-1$ to $1$ .

Calculation:

The remaining part can be calculated as,

For the proportion of the population lying within $2$ standard deviation is,

  $\underset{-3}{\overset{3}{{\displaystyle \int }}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}dx\approx 0.9545$ .

Again,

For the proportion of the population lying within $3$ standard deviation is,

  $\underset{-3}{\overset{3}{{\displaystyle \int }}}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}dx\approx 0.997$ .