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 use​Chebyshev'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/k² ] * 100% of the data fall within k standard deviations of the mean.

a) For k = 3

[ 1 - (1/3²) ] * 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 - (1/4²) ] * 100% = 0.9375 * 100% = 93.75 %

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