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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 $123 and a standard deviation of $9. If nothing is known about the shape of the distribution, what percentage of homes will have a monthly utility bill of less than $105

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Answer:

At most 25% of homes will have a monthly utility bill of less than $105.

Step-by-step explanation:

Chebyshev Theorem

The Chebyshev Theorem can also be applied to non-normal distribution. It states that:

At least 75% of the measures are within 2 standard deviations of the mean.

At least 89% of the measures are within 3 standard deviations of the mean.

An in general terms, the percentage of measures within k standard deviations of the mean is given by [tex]100(1 - \frac{1}{k^{2}})[/tex].

In this question, we have that:

The mean is $123.

The standard deviation is $9.

What percentage of homes will have a monthly utility bill of less than $105

105 = 123 - 2*9

So 105 is two standard deviations below the mean.

We have that, by the Chebyshev Theorem, at least 75% of the measures will be within 2 standard deviations from the mean, so at most 25% of the measures will be more than two standard deviations of the mean(greater than 141 ou lesser than 105).

So at most 25% of homes will have a monthly utility bill of less than $105