Question

Suppose that prices of a gallon of milk at various stores in one town have a mean of $3.96 with a standard deviation of $0.12. Using Chebyshev's Theorem, what is the minimum percentage of stores that sell a gallon of milk for between $3.72 and $4.20? Round your answer to one decimal place.

a) 88.9%
b) 75.0%
c) 62.5%
d) 50.0%

Answer 1

Using Chebyshev's Theorem, the minimum percentage of stores selling milk between $3.72 and $4.20, which is 2 standard deviations from the mean, is 75.0%.

Step-by-step explanation:

The student is asking about an application of Chebyshev's Theorem in the context of determining the minimum percentage of stores selling milk within a certain price range based on the given mean and standard deviation. Chebyshev's Theorem can be used for any distribution shape to estimate the minimum proportion of observations that fall within a certain number of standard deviations from the mean.

In this case, the price range from $3.72 to $4.20 constitutes 2 standard deviations below and above the mean ($3.96) since the standard deviation is $0.12. Applying Chebyshev's Theorem, the minimum percentage of stores that would fall within this range is 1 - (1/k^2), where k is the number of standard deviations, which is 2 in this case. Therefore, the minimum percentage is 1 - (1/2^2) = 1 - 1/4 = 0.75 or 75%. Hence, the correct answer is (b) 75.0%