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Chapter 4: Problem 51

Use a graphing utility to graph the normal probability density function with \(\mu=0\) and \(\sigma=2,3,\) and 4 in the same viewing window. What effect does the standard deviation \(\sigma\) have on the function? Explain your reasoning.

Short Answer

Expert verified
Increasing the standard deviation, \( \sigma \), causes the graph of the normal probability density function to widen, indicating a larger spread of data. Conversely, decreasing \( \sigma \) causes the graph to become narrower, indicating a smaller spread of data.

Step by step solution

01

Understanding the Normal Density Function

The normal probability density function is a bell-shaped curve described by the equation: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} ({\frac{x-\mu}{\sigma}})^2}\] where \( \mu \) is the mean and \( \sigma \) is the standard deviation. The standard deviation, \( \sigma \), determines how spread out the distribution is. If \( \sigma \) increases, the graph will become wider (more spread out) and if \( \sigma \) decreases, the graph will become narrower (less spread out).
02

Graphing the Function with Different \( \sigma \) Values

Using a graphing utility, insert the normal density function equation using \( \mu = 0 \) and \( \sigma = 2 \), then 3, and finally 4. Plot all three graphs on the same viewing window for better comparison.
03

Observing the Effect of \( \sigma \)

Observe the graphs. As \( \sigma \) increases, you will notice that the graph becomes wider, indicating a larger spread of data. Conversely, a smaller \( \sigma \) results in a narrower graph indicating a smaller spread of data. This reflects the principle that a higher standard deviation results in a wider distribution of data, and vice versa.

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