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%

Step-by-step 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 $(1-\dfrac{1}{k^2})$ of the values  lies with in $k\sigma$ from mean.

For k= 3

At-least $(1-\dfrac{1}{3^2})$ of the values lies within $3\sigma$ from mean.

$1-\dfrac{1}{3^2}=1-\dfrac{1}{9}=\dfrac{8}{9}$

In percent = $\dfrac{8}{9}\times100\%\approx88.89\%$

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