Question

A recent survey was conducted to compare the cost of solar energy to the cost of gas or electric energy. Results of the survey revealed that the distribution of the amount of the monthly utility bill of a 3-bedroom house using gas or electric energy had a mean of $93 and a standard deviation of $11. Three solar homes reported monthly utility bills of $56, $54, and $51. Which of the following statements is true?

A. Homes using solar power always have lower utility bills than homes using only gas and electricity.
B. Homes using solar power may have lower utility bills than homes using only gas and electricity.
C. Homes using solar power may actually have higher utility bills than homes using only gas and electricity.
D. The utility bills for homes using solar power are about the same as those for homes using only gas and electricity.

Answer 1

A. Homes using solar power always have lower utility bills than homes using only gas and electricity.

Explanation:

Gas and Electricity

`\bar x = 93`

`\sigma = 11`

Solar

Bills: 56, 54 and 51

Required

Determine which of the statements is true

First, we calculate the mean of the given data

`\bar x = (\sum x)/(n)`

`\bar x = (56 + 54 + 51)/(3)`

`\bar x = (161)/(3)`

`\bar x = 53.67`

Then, the standard deviation

`\sigma = \sqrt{(\sum(x - x_i)^2)/(n)}`

`\sigma = \sqrt{((56 - 53.67)^2 + (54 - 53.67)^2 + (51 - 53.67)^2)/(3)}`

`\sigma = \sqrt{(12.67)/(3)}`

`\sigma = √(4.223)`

`\sigma = 2.05`

So, we have:

Gas or Electricity

`\bar x = 93`

`\sigma = 11`

Solar

`\bar x = 53.67`

`\sigma = 2.05`

By comparison.

The mean and standard deviation of homes using solar panel is less than home using gas and electricity.

This implies that: (a) is true.