Suppose, household color TVs are replaced at an average age of ? = 7.8 years after purchase, and the (95% of data) range was from 3.8 to 11.8 years. Thus, the range was 11.8 − 3.8 = 8.0 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 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 ? − 2? to ? + 2? is often used for "commonly occurring" data values. Note that the interval from ? − 2? to ? + 2? is 4? 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. (Round your answer to one decimal place.) yrs (b) What is the probability that someone will keep a color TV more than 5 years before replacement? (Round your answer to four decimal places.) (c) What is the probability that someone will keep a color TV fewer than 10 years before replacement? (Round your answer to four decimal places.) (d) Assume that the average life of a color TV is 7.8 years with a standard deviation of 2.0 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 6% of the TVs under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)? yrs

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose, household color TVs are replaced at an average age of ? = 7.8 years after purchase, and the (95% of data) range was from 3.8 to 11.8 years. Thus, the range was 11.8 − 3.8 = 8.0 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 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 ? − 2? to ? + 2? is often used for "commonly occurring" data values. Note that the interval from ? − 2? to ? + 2? is 4? 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

≈

high value − low value

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. (Round your answer to one decimal place.)
yrs

(b) What is the probability that someone will keep a color TV more than 5 years before replacement? (Round your answer to four decimal places.)

(c) What is the probability that someone will keep a color TV fewer than 10 years before replacement? (Round your answer to four decimal places.)

(d) Assume that the average life of a color TV is 7.8 years with a standard deviation of 2.0 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 6% of the TVs under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)?
yrs

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