At one point the average price of regular unleaded gasoline was $3.38 per gallon

Question

At one point the average price of regular unleaded gasoline was
$3.38
per gallon. Assume that the standard deviation price per gallon is
$0.08
per gallon and useChebyshev's inequality to answer the following.
(a) What percentage of gasoline stations had prices within
22
standard deviations of the mean?
(b) What percentage of gasoline stations had prices within
1.5
standard deviations of the mean? What are the gasoline prices that are within
1.5
standard deviations of the mean?
(c) What is the minimum percentage of gasoline stations that had prices between
$3.14
and
$3.62

Explanation / Answer

Here we are given that the mean = $3.38 and the standard deviation, SD = $0.08

According to the Chebyshev's inequality, at least 11/k2 of the distribution's values are within k standard deviations of the mean

a) Here we are given that K = 22, therefore 11/k2 = 1- 1/222 = 0.9979

Therefore at least 99.79% of the observations must lie within 22 standard deviations of the mean.

b) Here we have K =1.5 Therefore, 11/k2 = 1- 1/1.52 = 0.5556

Therefore at least 55.56% of the observations must lie within 1.5 standard deviations of the mean.

The scores here would be: Mean - K*SD = $3.38 - 1.5*0.08 = $3.26 and Mean + K*SD = $3.38 + 1.5*0.08 = $3.50

Therefore between $3.26 and $3.50

c) 3.14 = 3.38 - 3*0.08 and 3.62 = 3.38 + 3*0.08

Therefore K = 3 Here,

11/k2 = 1 - 1/9 = 0.8889

Therefore at least 88.89% of the observations lies in that interval.

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