At one point the price of regular unleaded gasoline was $3.39 per gallon. Assume

Question

At one point the price of regular unleaded gasoline was $3.39 per gallon. Assume the standard deviation price per gallon is 0.07 per gallon and use Chebyshev's inequality to answer the following.

(a) What percentage of gasoline stations had prices within

4 standard deviations of the mean?

(b) What percentage of gasoline stations had prices within

2.5 standard deviations of the mean?

(c) The gas prices that are within 2.5 standard deviations are ? to ?

(d)What minimum percent of gas stations had prices between 3.25 and 3.53?

Explanation / Answer

Here mean = 3.39, standard deviation = 0.07
Chebyshev's inequality theorem states that no more than 1/k2 of the distribution's values can be more than k standard deviations away from the mean

a) Here k=4, so 1/k^2 = 1/16 = 0.0625 = 6.25%. So (100 - 6.25) = 93.75% of the gasoline stations had prices within 4 standard deviations of the mean.

b) Here k=2.5, so 1/k^2 = 1/6.25 = 0.16 = 16%. So (100 - 16) = 84% of the gasoline stations had prices within 2.5 standard deviations of the mean.

c) Gas prices within 2.5 standard devaiations are - (3.39 - 2.5*0.07) to (3.39 + 2.5*0.07)
= 3.215 to 3.565

d) standard deviaitons corresponding to a difference between 3.53 and 3.25 = (3.53 - 3.25 )/0.07 = 4
So this corresponds to +- 2 standard deviations.

So here k = 2, so 1/k^2 = 1/4 = 0.25 = 25%. So (100 - 25) = 75% of the gasoline stations had prices between 3.25 and 3.53.

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