Suppose, household color TVs are replaced at an average age of = 8.2 years after purchase, and the (95% of data) range was from

Suppose, household color TVs are replaced at an average age of μ = 8.2 years after purchase, and the (95% of data) range was from 6.2 to 10.2 years. Thus, the range was 10.2 – 6.2 = 4.0 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 indicates that for a symmetrical 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,

standard deviation ≈

range 4

≈

high value – 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) Assume that the average life of a color TV is 8.2 years with a standard deviation of 1.7 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 15% of the TVs under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)?