Question

In December 2018, the average price of regular unleaded gasoline excluding taxes in the United States was $3.06 per gallon
Assume that the standard deviation price per gallon is $0.05 per gallon and use Chebyshev's Inequality to answer the following

(a) What minimum percentage of gasoline stations had prices within 2 standard deviations of the mean?

(b) What minimum percentage of gasoline stations had prices within 2.5 standard deviations of the mean?

(c) What is the minimum percentage of gasoline stations that had prices between $2.91 and $3.21?

Answer 1

Using Chebyshev's Theorem, it is found that:

a) The minimum percentage is 75%.

b) The minimum percentage is 84%.

c) The minimum percentage is 89%.

By Chebyshev's Theorem, the minimum percentage of measures within k standard deviations of the mean is:

$P = 100\left(1 - \frac{1}{k^{2}}\right)$

Item a:

Within 2 standard deviations, hence k = 2, and:

$P = 100\left(1 - \frac{1}{2^{2}}\right) = 100\left(\frac{3}{4}\right) = 75$

The minimum percentage is 75%.

Item b:

2.5 standard deviations, hence k = 2.5, and:

$P = 100\left(1 - \frac{1}{2.5^{2}}\right) = 84$

The minimum percentage is 84%.

Item c:

3 standard deviations, hence k = 3, and:

$P = 100\left(1 - \frac{1}{3^{2}}\right) = 89$

The minimum percentage is 89%.

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