Chapter 6.3, Problem 31P

Expand Your Knowledge: Estimating the Standard Deviation Consumer Reports gave information about the ages at which various household products are replaced. For example, color TVs are replaced at an average age of μ = 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.

1. (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 μ – 2σ to μ + 2σ is often used for “commonly occurring” data values. Note that the interval from μ − 2σ to μ + 2σ is 4σ 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,

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

   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.

2. (b) What is the probability that someone will keep a color TV more than 5 years before replacement?
3. (c) What is the probability that someone will keep a color TV fewer than 10 years before replacement?
4. (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)?