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The question asks for the percentage of homes that will have a monthly utility bill between $91 and $143, given the following information:

- The monthly utility bill has a **mean** of $117.
- The **standard deviation** is $13.
- The distribution is **normal**.

We can use the properties of the normal distribution to solve this. The first step is to standardize the values ($91 and $143) by converting them into **z-scores**, which represent the number of standard deviations a value is away from the mean.

The z-score formula is:

\[
z = \frac{x - \mu}{\sigma}
\]

where:
- $ x $ is the value (either $91 or $143),
- $ \mu $ is the mean ($117),
- $ \sigma $ is the standard deviation ($13).

Let's compute the z-scores for $91 and $143.The z-scores for $91 and $143 are:

- For $91: $ z = -2.0 $
- For $143: $ z = 2.0 $

Next, we need to find the percentage of values that fall between these two z-scores in a normal distribution. Using the standard normal distribution table:

- The cumulative probability for $ z = -2.0 $ is approximately 0.0228 (2.28%).
- The cumulative probability for $ z = 2.0 $ is approximately 0.9772 (97.72%).

To find the percentage of homes with a monthly utility bill between $91 and $143, subtract the two probabilities:

\[
\text{Percentage} = 0.9772 - 0.0228 = 0.9544
\]

So, approximately **95.44%** of homes will have a monthly utility bill between $91 and $143.

Would you like further details or have any other questions?

Here are 5 related questions to deepen understanding:
1. How do z-scores help in interpreting normal distributions?
2. What is the empirical rule for a normal distribution?
3. How would the percentage change if the standard deviation were larger?
4. What are some other ways to calculate areas under the normal curve?
5. How would the interpretation change if the distribution were not normal?

**Tip:** Understanding the relationship between z-scores and probabilities can be very useful in making statistical predictions about data.

Solar energy is considered by many to be the energy of the future. A recent survey was taken 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 three-bedroom house using gas or electric energy had a mean of $117 and a standard deviation of $13. If the distribution can be considered normal: Approximately what percentage of homes will have a monthly utility bill between $91 and $143?

Learn how to calculate the percentage of homes with a monthly utility bill between $91 and $143 using the properties of a normal distribution. The problem involves converting values to z-scores and using the cumulative probability to find the desired percentage. Includes detailed steps and explanations.

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