Suppose, household color TVs are replaced at an average age of μ = 7.4 years after purchase, and the (95% of data) range was from 5.4 to 9.4 years. Thus, the range was 9.4 − 5.4 = 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 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 deviationFor 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 7.4 years with a standard deviation of 1.0 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 14% of the TVs under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)? yrs
Suppose, household color TVs are replaced at an average age of μ = 7.4 years after purchase, and the (95% of data) range was from 5.4 to 9.4 years. Thus, the range was 9.4 − 5.4 = 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 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,
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 7.4 years with a standard deviation of 1.0 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 14% 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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