Question

Suppose that prices of a gallon of milk at various stores in one town have a mean of $3.73$⁢3.73 with a standard deviation of $0.10$⁢0.10. Using Chebyshev's Theorem, what is the minimum percentage of stores that sell a gallon of milk for between $3.43$⁢3.43 and $4.03$⁢4.03? Round your answer to one decimal place.

Answer 1

Answer: At-least 88.89%

Explanation:

As per given , we have

Population mean :
`\mu=\$3.73`

Standard deviation :
`\sigma=\$0.10`

Now , $3.43= $⁢3.73- 3(0.10) =
`\mu-3\sigma`

$⁢4.03 = $⁢3.73+3(0.10) =
`\mu+3\sigma`

i.e. $3.43 is 3 standard deviations below mean and $⁢4.03 is 3 standard deviations above mean .

To find : the minimum percentage of stores that sell a gallon of milk for between $3.43 and $4.03.

i.e. to find minimum percentage of stores that sell a gallon of milk lies within 3 standard deviations from mean.

According to Chebyshev, At-least
`(1-(1)/(k^2))` of the values lies with in
`k\sigma` from mean.

For k= 3

At-least
`(1-(1)/(3^2))` of the values lies within
`3\sigma` from mean.

`1-(1)/(3^2)=1-(1)/(9)=(8)/(9)`

In percent =
`(8)/(9)*100\%\approx88.89\%`

Hence, the minimum percentage of stores that sell a gallon of milk for between $3.43 and $4.03 = At-least 88.89%