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The question asks to estimate the percentage of showers that used water within specific ranges, assuming a normal distribution of water usage with an average (mean) of 16.7 gallons and a standard deviation of 2.9 gallons. The image provides the percentage values for parts (a) through (c), and part (d) requires calculation.

Let's break down the steps to solve part (d) — "between 8 and 22.5 gallons."

### Steps:
1. **Convert raw values to z-scores**:
   The z-score formula is:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   where:
   - \( X \) is the data point (the amount of water),
   - \( \mu = 16.7 \) (the mean),
   - \( \sigma = 2.9 \) (the standard deviation).

2. **Z-scores for the given values**:
   - For 8 gallons: 
     \[
     Z = \frac{8 - 16.7}{2.9} \approx -3.00
     \]
   - For 22.5 gallons:
     \[
     Z = \frac{22.5 - 16.7}{2.9} \approx 2.00
     \]

3. **Find the cumulative probability**:
   Using standard normal distribution tables or a calculator:
   - The cumulative probability for \( Z = -3.00 \) is approximately 0.0013 (or 0.13%).
   - The cumulative probability for \( Z = 2.00 \) is approximately 0.9772 (or 97.72%).

4. **Calculate the percentage**:
   The percentage of showers between 8 and 22.5 gallons is:
   \[
   97.72\% - 0.13\% = 97.59\%
   \]

Therefore, the percentage for part (d) is **97.6%**.

Would you like more details or have any questions about this? Here are 5 related questions for practice:

1. How would you calculate the percentage of showers using less than 8 gallons?
2. What is the percentage of showers using more than 22.5 gallons?
3. If the mean water usage increased to 18 gallons, how would this affect the percentages?
4. How would you calculate the percentage for a range that is not symmetric around the mean?
5. What is the probability of using exactly 16.7 gallons of water?

**Tip**: Z-scores help you convert raw values into a common scale, making it easier to compare different data points using a normal distribution.

Suppose that water usages in American showers are normally distributed, with an average shower using 16.7 gallons, and a standard deviation of 2.9 gallons. Estimate the percentage of showers that used:
(a) between 10.9 and 22.5 gallons.
(b) more than 25.4 gallons.
(c) less than 10.9 gallons.
(d) between 8 and 22.5 gallons.

Learn how to estimate water usage in American showers, which are normally distributed, with an average of 16.7 gallons and a standard deviation of 2.9 gallons. Calculate the percentage of showers using between 8 and 22.5 gallons, using more than 25.4 gallons, and more. Suitable for high school and college statistics students.

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