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The problem asks to estimate percentages of showers that use certain amounts of water, assuming that water usage is normally distributed with:

- Mean \(\mu = 17.4\) gallons
- Standard deviation \(\sigma = 2.4\) gallons

The key here is to calculate the **Z-scores** for each question and then use the standard normal distribution to find the corresponding percentages.

### (a) Between 12.6 and 22.2 gallons:

1. \( Z_1 = \frac{12.6 - 17.4}{2.4} = \frac{-4.8}{2.4} = -2 \)
2. \( Z_2 = \frac{22.2 - 17.4}{2.4} = \frac{4.8}{2.4} = 2 \)

We now look up the cumulative probabilities for these Z-scores.

### (b) More than 24.6 gallons:

1. \( Z = \frac{24.6 - 17.4}{2.4} = \frac{7.2}{2.4} = 3 \)

### (c) Less than 15 gallons:

1. \( Z = \frac{15 - 17.4}{2.4} = \frac{-2.4}{2.4} = -1 \)

### (d) Between 15 and 22.2 gallons:

1. \( Z_1 = \frac{15 - 17.4}{2.4} = \frac{-2.4}{2.4} = -1 \)
2. \( Z_2 = \frac{22.2 - 17.4}{2.4} = 2 \)

Let me compute the corresponding probabilities for these Z-scores.Here are the estimated percentages for each part of the question:

(a) **Between 12.6 and 22.2 gallons**: 95.45%

(b) **More than 24.6 gallons**: 0.13%

(c) **Less than 15 gallons**: 15.87%

(d) **Between 15 and 22.2 gallons**: 81.86%

Would you like more details or clarification on any of these calculations?

Here are 5 related questions you might find interesting:
1. How do Z-scores relate to the normal distribution curve?
2. What is the probability of using less than 10 gallons of water?
3. Can we estimate the median water usage with this data?
4. How would the percentages change if the standard deviation were different?
5. What is the significance of using the cumulative distribution function (CDF) in normal distributions?

**Tip**: Z-scores tell us how far a value is from the mean in terms of standard deviations, which is key to using normal distributions!

Suppose that water usages in American showers are normally distributed, with an average shower using 17.4 gallons, and a standard deviation of 2.4 gallons. Estimate the percentage of showers that used (a) between 12.6 and 22.2 gallons, (b) more than 24.6 gallons, (c) less than 15 gallons, (d) between 15 and 22.2 gallons.

Learn how to estimate percentages of showers based on water usage with a normal distribution. Given a mean of 17.4 gallons and a standard deviation of 2.4 gallons, this problem involves calculating Z-scores and finding corresponding probabilities for different ranges. Suitable for Grades 11-12 students studying normal distributions.

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