Question

At one point the average price of regular unleaded gasoline was 53.42 per gallon. Assume that the standard deviation price per gallon is 50.04 per gallon and use Chebyshev's inequality to answer the following. (a) What percentage of gasoline stations had prices within 3 standard deviations of the mean? (b) What percentage of gasoline stations had prices within 2.5 standard deviations of the mean? What are the gasoline prices that are within 2.5 standard deviations of the mean? (c) What is the minimum percentage of gasoline stations that had prices between $3.34 and $3.50?

Answer 1

Average price of regular unleaded gasoline = $\mu$ = $ 3.42 per gallon

standard deviation price per gallon = $\sigma$ = $ 0.04 per gallon

Here as per Chebyshev's inequality

Pr(l X - $\mu$ l   $\geq$ k $\sigma$ )   $\leq$ 1/k²

(a) Here the percentage of gasoline stations had prices withing 3 standard deviation away = 1 - 1/k² = 1- 1/3² = 1/9 = 11.11%

(b) Percentage of gasoline standard had prices within 2.5 standard deviation of the mean = 1 - 1/k² = 1- 1/2.5² = 0.84

(c) Here

Pr($ 3.34 < X < $ 3.50) = Pr( 3.42 - 2 * 0.04 < X < 3.42 + 2 * 0.04) = 1 - 1/k² = 1- 1/2² = 0.75 = 75%