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To solve the problem described in the image, we need to find the percentage of homes that have a monthly utility bill less than $78, given that the distribution of monthly utility bills is normally distributed with:

- Mean ($\mu$) = $117
- Standard deviation ($\sigma$) = $13

The question asks for the percentage of homes with a utility bill less than $78. To solve this, we will use the Z-score formula for a normal distribution:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- $X$ is the value of interest ($78),
- $\mu$ is the mean ($117),
- $\sigma$ is the standard deviation ($13).

Let me calculate this Z-score first.The Z-score for a monthly utility bill of $78 is $-3.0$. 

Now, we can use the standard normal distribution table or a cumulative distribution function (CDF) to find the probability associated with this Z-score. A Z-score of $-3.0$ corresponds to a very low percentage, indicating that very few homes will have a utility bill below $78.

Let me calculate the cumulative probability for this Z-score.Approximately 0.135% of homes will have a monthly utility bill less than $78.

Would you like more details or have any questions about this calculation?

Here are 5 related questions for further exploration:
1. What percentage of homes have a utility bill greater than $150?
2. How can we calculate the percentage of homes with utility bills between $100 and $130?
3. If the standard deviation increases to $20, how does that affect the percentage of homes with a bill below $78?
4. How do we interpret a Z-score of -3.0 in practical terms?
5. What is the probability of a home having a utility bill exactly equal to the mean of $117?

**Tip**: Understanding how to read Z-scores and their corresponding probabilities from a normal distribution table is a fundamental skill in statistics.

Approximately what percentage of homes will have a monthly utility bill less than $78?

Learn how to calculate the percentage of homes with a monthly utility bill less than $78 using the Z-score formula and the properties of the normal distribution. This problem involves key statistical concepts such as the mean, standard deviation, and the Z-score, making it ideal for high school students in Grades 9-12.

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