Chapter 7.3, Problem 31P

Expand Your Knowledge: Estimating the Standard DeviationConsumer Reports gave information about the ages at which various household products are replaced. For example, color TVs are replaced at an average age of $\mu =8$ years after purchase, and the ( $95\%$ of data) range was from 5 to 11 years. Thus, the range was $11–5=6$ years. Let x be the age (in years) at which a color TV is replaced. Assume that x has a distribution that is approximately normal.

(a) The empirical rule (see Section 7.1) indicates that for a symmetric and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a $95\%$ range of data values extending from $\mu -2\sigma \text{ to }\mu +2\sigma$ is often used for “commonly occurring” data values. Note that the interval from $\mu -2\sigma \text{ to }\mu +2\sigma \text{ is }4\sigma$ in length. This leads to a “rule of thumb" for estimating the standard deviation from a $95\%$ range of data values.

ESTIMATING THE STANDARD DEVIATION

For a symmetric, bell-shaped distribution,

$\text{standard deviation}\approx \frac{\text{range}}{4}\approx \frac{\text{high value}-\text{low value}}{4}$

where it is estimated that about 95% of the commonly occurring data values fall into this range.

Use this “rule of thumb" to approximate the standard deviation of x values, where x is the age (in years) at which a color TV is replaced.

(b) What is the probability that someone will keep a color TV more than 5 years before replacement?

(c) What is the probability that someone will keep a color TV fewer than 10 years before replacement?

(d)Inverse Normal Distribution Assume that the average life of a color TV is 8 years with a standard deviation of 1.5 years before it breaks. Suppose that a company guarantees color TVs and will replace a TV that breaks while under guarantee with a new one. However, the company does not want to replace more than 10% of the TVs under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)?