Suppose, household color TVs are replaced at an average age of \mu

Paganellash

Paganellash

Answered question

2021-11-27

Suppose, household color TVs are replaced at an average age of μ=8.4years after purchase, and the (95% of data) range was from 4.4 to 12.4 years. Thus, the range was 12.44.4=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 μ2σ  μ+2σ is often used for "commonly occurring" data values. Note that the interval from μ2σ  μ+2σis4σ 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 deviationran 4high valuelow 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.)
(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.)

Answer & Explanation

Ceitheart

Ceitheart

Beginner2021-11-28Added 14 answers

Solution:
If a random variable x has a distribution with mean μ and standard deviation σ, then the z-score is defined as,
zscore=xμσN(0,1)
Step 3
(a)Estimating the standard deviation:
It is given that, for a symmetric or bell shaped distribution, standard deviation is,
Standard deviationRange 4.
It is given that, range is, 8.0 years.
That is, the standard deviation is 2 (=84).
Thus, using the rule of thumb, the approximate standard deviation of x values is 2 years.
Step 4
(b)The probability that someone will keep a colour TV more than 5 years before replacement:
It is given that the household colour TVs are replaced at an average of μ=8.4 years after purchase. It is found that, standard deviation, σ=2years.
The random variable x represents the age at which a colour TV is replaced.
The probability that the someone will keep a colour TV more than 5 years before replacement is 0.9554 and is obtained as shown below:
P(x>5)=P(xμσ>58.42)
=P(z1.7)
=1P(z1.7)
[EXCEL formula: NORM.DIST (-1.7,0,1,TRUE)]
=10.0446
=0.9554
Step 5
Thus, the probability that someone will keep a colour TV more than 5 years before replacement is 0.9554.
(c)The probability that someone will keep a colour TV fewer than 10 years before replacement:
The probability that someone will keep a colour TV fewer than 10 years before replacement is 0.7881 and is obtained as shown below:
P(x<10)=P(xμσ<108.42)
=P(z<0.8)
=0.7881
[EXCEL formula: NORM.DIST (0.8,0,1,TRUE)]
Thus, the probability that someone will keep a colour TV fewer than 10 years before replacement is 0.7881.
Step 6
Answer:
(a)The approximate standard deviation of x values is 2 years.
(b)The probability that someone will keep a colour TV more than 5 years before replacement is 0.9554.
(c)The probability that someone will keep a colour TV fewer than 10 years before replacement is 0.7881.

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