Chapter 11, Problem 19CT

Solution

To plot: A normal distribution curve showing one, two, and three standard deviations from the mean.

Given:

A data set is normally distributed with a mean of 37 and a standard deviation of 8.1.

Concept used:

The central limit theorem state that in a normal distribution within one standard deviation of the mean, there is about 68% of the data, within two standard deviations of the mean, there is about 95% of the data, and within three standard deviations of the mean, there is about 99,7% of the data.

Calculation:

The mean and standard deviation of the normal distribution are 37 and 8.1 respectively. One, two, and three standard deviations of the mean data points are calculated as shown below

  $\begin{array}{l}\mu -\sigma =37-8.1=28.9\\ \mu +\sigma =37+8.1=45.1\\ \mu -2\sigma =37-2\times 8.1=20.8\\ \mu +2\sigma =37+2\times 8.1=53.2\\ \mu -3\sigma =37-3\times 8.1=12.7\\ \mu +3\sigma =37+3\times 8.1=61.3\end{array}$

Plot:

A normal distribution curve is a bell-shaped curve with its symmetrical about the mean or center and is plotted as shown below