https://file.aiquickdraw.com/mathbot/file/1725853898799pkp2wrqg.jpg

The problem in the image provides the following information:

- A survey was taken to compare the cost of solar energy to gas/electric energy.
- The distribution of the monthly utility bill for a three-bedroom house using gas or electric energy has:
  - A mean ($\mu$) of $117
  - A standard deviation ($\sigma$) of $13.
- The question asks what percentage of homes will have a monthly utility bill of more than $104.

### Solution:

This is a normal distribution problem. To find the percentage of homes with a bill greater than $104, we need to standardize the value and use the **z-score formula**:

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

Where:
- $x = 104$ (the value we are comparing)
- $\mu = 117$ (the mean)
- $\sigma = 13$ (the standard deviation)

Now, let's calculate the z-score.The z-score is $-1.0$. 

To find the percentage of homes with a utility bill greater than $104, we need to calculate the area to the right of $z = -1.0$ on the standard normal distribution curve.

Using standard normal distribution tables or a calculator, the area to the left of $z = -1.0$ is approximately 0.1587, which means about 15.87% of homes have a bill less than $104.

Therefore, the percentage of homes with a bill greater than $104 is:

\[
1 - 0.1587 = 0.8413 \text{ or } 84.13\%
\]

So, approximately 84.13% of homes have a monthly utility bill greater than $104.

Would you like further details or clarifications?

### Related Questions:
1. How would the percentage change if the bill was $130 instead of $104?
2. Can you explain how the z-score helps in solving normal distribution problems?
3. What is the probability that a bill falls between $104 and $117?
4. How does increasing the standard deviation affect the percentage of homes with bills over $104?
5. What would the z-score be for a bill of $150?

**Tip:** Using the z-score allows you to compare values from different normal distributions easily.

Approximately what percentage of homes will have a monthly utility bill of more than $104?

Solve a statistics problem involving the normal distribution of monthly utility bills with a mean of $117 and a standard deviation of $13. Learn how to calculate the percentage of homes with a bill higher than $104 using the Z-Score formula. Suitable for students in Grades 10-12.

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