Question

At one point the average price of regular unleaded gasoline was $3.58 per gallon. Assume that the standard deviation price per gallon is $0.05 per gallon and useChebyshev's inequality to answer the following.

(a) What percentage of gasoline stations had prices within 33 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 $3.38 and $3.78?

Answer #1

Given that, mean = $3.58 per gallon

standard deviation = $0.05 per gallon

According to Chebyshev's inequality,

at least [ 1 - (1/k^{2} ] * 100% of the data fall within
k standard deviations of the mean.

a) For k = 3

[ 1 - (1/3^{2}) ] * 100% = 0.8889 * 100% = 88.89%

Therefore, **88.89 %** of gasoline stations had
prices within 3 standard deviations of the mean.

b) For k = 1.5

[ 1 - (1/(1.5)^{2}) ] * 100% = 0.5556 * 100% =
55.56%

Therefore, **55.56** % of gasoline stations had
prices within 1.5 standard deviations of the mean.

And

3.58 - (1.5 * 005) = 3.58 - 0.075 = 3.505 And

3.58 + (1.5 * 0.05) = 3.58 + 0.075 = 3.655

Therefore, 55.56% of the gasoline stations had prices between
**$ 3.505** and **$ 3.655** per
gallon.

c) For k = 4

3.58 - (4 * 0.05) = 3.58 - 0.20 = 3.38 And

3.58 + (4 * 0.05) = 3.58 + 0.20 = 3.78

And

[ 1 - (1/4^{2}) ] * 100% = 0.9375 * 100% = 93.75 %

Therefore, minimum percentage of gasoline stations that had
prices between $ 3.38 and $ 3.78 is **93.75 %**

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