Question

In December 2014, the average price of regular unleaded gasoline excluding taxes in the United States was $3.06 per gallon, according to the Energy Information Administration. Assume that the standard deviation price per gallon is $0.06 per gallon to answer the following.(a) What minimum percentage of gasoline stations had prices within 3 standard deviations of the mean? (b) What minimum 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 $2.94 and $3.18?

Answer 1

Answer:

a) 88.89%

b) 84%

c) 75%

Step-by-step explanation:

We are given the following in the question:

Average price = $3.06 per gallon

Standard deviation =  $0.06 per gallon

Chebyshev's Rule:

• According to this rule atleast $1 - \dfrac{1}{k^2}$  percent of data lies within k standard deviation of mean.

a) minimum percentage of gasoline stations had prices within 3 standard deviations of the mean

We put k = 3

$1 - \dfrac{1}{(3)^2} = 0.8889 = 88.89\%$

Thus, minimum 88.89% of gasoline stations had prices within 3 standard deviations of the mean.

b) percentage of gasoline stations had prices within 2.5 standard deviations of the mean

We put k = 2.5

$1 - \dfrac{1}{(2.5)^2} = 0.84 = 84\%$

Thus, minimum 84% of gasoline stations had prices within 2.5 standard deviations of the mean.

Range of gasoline price:

$\mu - 2.5(\sigma) = 3.06 - 2.5(0.06) =2.91\\\mu + 2.5(\sigma) = 3.06 + 2.5(0.06) =3.21$

c) minimum percentage of gasoline stations that had prices between $2.94 and $3.18

We can express $2.94 and $3.18 as:

$2.94 = 3.06 - 2(0.06) = \mu - 2(\sigma)\\3.18 =3.06 + 2(0.06) = \mu + 2(\sigma)$

We put k = 2

$1 - \dfrac{1}{(2)^2} = 0.75 = 75\%$

Thus, minimum 75% of gasoline stations had prices between $2.94 and $3.18.