Chapter 15.8, Problem 7P

By computer find the value of the normal cumulative distribution function at $\mu +\sigma$, $\mu +2\sigma ,\mu +3\sigma ,$ and satisfy yourself that these are independent of your choices for $\mu$ and $\sigma .$ Find the probabilities that $x$ is within 1, 2, or 3 standard deviations of its mean value $\mu$ to verify the results stated in the paragraph following $(8.2).$ Hint: See Figure (8.1). The probability that $x$ is within 1 standard deviation of its mean value is the area from $\mu -\sigma$ to $\mu +\sigma$; this is twice the area from $\mu$ to $\mu +\sigma .$ Subtract $\frac{1}{2}(\text{ that is the area from }-\infty \text{ to }\mu \text{ ) from your value of }F(\mu +\sigma )$ and then double the result.