Chapter 2.2, Problem 48E

To determine

Represent the Normal density curve with mean and the points that are 1, 2, and 3 standard deviations from the mean.

Explanation of Solution

Given:

Mean, $\mu =0.261$

Standard deviation, $\sigma =0.034$

According to 68 − 95 − 99.7 rule:

68% of the data of a normal distribution lies with 1 standard deviation from the mean.

95% of the data of a normal distribution lies with 2 standard deviation from the mean.

99.7% of the data of a normal distribution lies with 1 standard deviation from the mean.

Then

The general Normal density graph is represented as:

  

Now,

Find the standard values on both sides of the mean.

From left to the mean,

3 standard deviation:

  $\mu -3\sigma =0.261-3\left(0.034\right)=0.159$

2 standard deviation:

  $\mu -2\sigma =0.261-2\left(0.034\right)=0.193$

1 standard deviation:

  $\mu -\sigma =0.261-0.034=0.227$

From right to the mean,

1 standard deviation:

  $\mu +\sigma =0.261+0.034=0.295$

2 standard deviation:

  $\mu +2\sigma =0.261+2\left(0.034\right)=0.329$

3 standard deviation:

  $\mu +3\sigma =0.261+3\left(0.034\right)=0.363$

Write the calculated values in the sketch below to represent the Normal density curve: