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The problem provided involves a normal distribution of monthly utility bills for three-bedroom houses, where the mean is $117 and the standard deviation is $13. We are asked to find the percentage of homes with monthly utility bills between $91 and $143.

### Step-by-step approach:

1. **Mean** ($\mu$): 117
2. **Standard Deviation** ($\sigma$): 13
3. **Range to find**: Between 91 and 143

We will first convert the given values into **z-scores** using the formula:

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

Where:
- $x$ is the value we are interested in,
- $\mu$ is the mean, and
- $\sigma$ is the standard deviation.

Let's now calculate the z-scores for $91 and $143:

- For $x = 91$:
  \[
  z_1 = \frac{91 - 117}{13}
  \]
  
- For $x = 143$:
  \[
  z_2 = \frac{143 - 117}{13}
  \]

I'll calculate the z-scores and the corresponding percentages.The z-scores for the utility bill values are:

- $ z_1 = -2.0 $ for $x = 91$
- $ z_2 = 2.0 $ for $x = 143$

Now, using the z-score table (standard normal distribution), we can find the percentage of values falling between these z-scores:

- The area to the left of $z = -2.0$ is approximately 2.28%.
- The area to the left of $z = 2.0$ is approximately 97.72%.

The percentage of homes with bills between $91 and $143 is the difference between these two areas:

\[
97.72\% - 2.28\% = 95.44\%
\]

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

Would you like further details or explanations? Here are some related questions:

1. What is a z-score, and how is it used in normal distribution problems?
2. How do we interpret areas under the normal curve?
3. What are the common uses of standard deviations in statistics?
4. How would the result change if the standard deviation were different?
5. What methods can be used to approximate percentages without a z-table?

**Tip**: For any normal distribution problem, knowing the 68-95-99.7 rule helps estimate percentages within 1, 2, or 3 standard deviations from the mean.

Approximately what percentage of homes will have a monthly utility bill between $91 and $143? The mean of the bills is $117, and the standard deviation is $13.

Learn how to calculate the percentage of homes with monthly utility bills between $91 and $143 using normal distribution. This problem involves a mean of $117, a standard deviation of $13, and includes key concepts like z-scores and the 68-95-99.7 rule.

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