  Suppose, household color TVs are replaced at an average age of μ = 8.4 years after purchase, and the (95% of data) range was from 4.4 to 12.4 years. Thus, the range was 12.4 − 4.4 = 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 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 ≈range4≈high value − low value4where 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 8.4 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)?

Question

Suppose, household color TVs are replaced at an average age of μ = 8.4 years after purchase, and the (95% of data) range was from 4.4 to 12.4 years. Thus, the range was 12.4 − 4.4 = 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 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 8.4 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)?
Step 1

Note:

Hi! Thank you for posting the question. Since your question has more than 3 parts, we have solved the first 3 parts for you. If you need help with any of the other parts, please re-post the question and mention the part you need help with.

Step 2

Solution:

If a random variable x has a distribution with mean µ and standard deviation σ, then the z-score is defined as,

Step 3

(a)Estimating the standard deviation:

It is given that, for a symmetric or bell shaped distribution, standard deviation is,

Standard deviation ≈ Range/4.

It is given that, range is, 8.0 years.

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