Estimating the Standard Deviation Consumer Reports gave information about the ages at which various household products are replaced. For example, TVs are replaced at an average age of m 5 8 years after purchase, and the (95% of data) range was from 5 to 11 years. Thus, the range was 112 5 5 6 years.

Let x be the age (in years) at which a TV is replaced.

Assume that x has a distribution that is approximately normal.

(a) The empirical rule (see Section 6.1) 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 m 2 2s to m 1 2s is often used for “commonly occurring”

data values. Note that the interval from m 2 2s to m 1 2s is 4s in length. This leads to a “rule of thumb” for estimating the standard deviation from a 95% range of data values.

Use this “rule of thumb” to approximate the standard deviation of x values, where x is the age

(in years) at which a TV is replaced.

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

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

(d) Inverse Normal Distribution Assume that the average life of a TV is 8 years with a standard deviation of 1.5 years before it breaks. Suppose that a company guarantees 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)?AppendixLO1

-20 95% 1 +20