Expand Your Knowledge: Estimating the Standard DeviationConsumer Reports gave information about the ages at which various household products are replaced. For example, color TVs are replaced at an average age of
μ
=
8
years after purchase, and the (
95
%
of data) range was from 5 to 11 years. Thus, the range was
11
–
5
=
6
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 (see Section 7.1) 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
−
2
σ
to
μ
+
2
σ
is often used for “commonly occurring” data values. Note that the interval from
μ
−
2
σ
to
μ
+
2
σ
is
4
σ
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)Inverse Normal Distribution Assume that the average life of a color TV is 8 years with a standard deviation of 1.5 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 10% of the TVs under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)?