Chapter 6.3, Problem 30E

Solution

a)

To construct: The three-given equations using same mean and different standard deviation.

Given information:

The equation of the normal curve is given.

Formula used:

The formula of the z-score is:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ \text{where}\\ x=\text{Row score}\\ \mu \text{=Population mean}\\ \sigma \text{=Population standard deviation}\end{array}$

Calculation:

Consider, mean is 2 and the standard deviation is 1.

The graph is:

  

Consider, mean is 2 and the standard deviation is 2.

The graph is:

  

Consider, mean is 2 and the standard deviation is 3.

The graph is:

  

b)

To discuss: The effect of the standard deviation on the normal curve.

From the above constructed graphs, it is clear that a high standard deviation suggests that the data is dispersed throughout a wide range of values, a low standard deviation suggests that the data points typically fall quite close to the mean.