Chapter 13, Problem 7PS

a.

To determine

To graph: Three equations of the given form where $\overline{x}$ is held constant and $\sigma$ varies.

Answer to Problem 7PS

Explanation of Solution

Given Information: Form is $\frac{1}{\sigma {\left(2\pi \right)}^{\frac{1}{2}}}{\left(e\right)}^{-\frac{{\left(x-\overline{x}\right)}^2}{2{\left(\sigma \right)}^2}}$ .

Graph:

  

Mean: $\overline{x}=25$

Red line represents the equation in which $\sigma =2$ , purple line represents the equation in which $\sigma =4$ , black line represents the equation in which $\sigma =6$ .

b.

To determine

To describe: The effect that standard deviation has on the shape of a normal curve.

Answer to Problem 7PS

The graph of the normal curve becomes wider and shorter.

Explanation of Solution

Given Information: Form is $\frac{1}{\sigma {\left(2\pi \right)}^{\frac{1}{2}}}{\left(e\right)}^{-\frac{{\left(x-\overline{x}\right)}^2}{2{\left(\sigma \right)}^2}}$ . Graph from previous part.

Interpretation:

As the standard deviation increases the graph of the normal curve becomes wider, shorter and gradually progresses towards flattening.