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The mean yearly rainfall in Sydney, Australia, is about 134 mm and the standard deviation is about 66 mm ("Annual maximums of," 2013). Assume rainfall is normally distributed. How many yearly mm of rainfall would there be in the top 15%? Round answer to 2 decimal places. Answer:

To find the amount of yearly rainfall that would be in the top 15%, we need to determine the corresponding rainfall value (in mm) that marks the 85th percentile of the normal distribution. Here's how we can solve this:

### Steps to Solve the Problem:

1. **Identify the parameters:**
   - Population mean (\(\mu\)) = 134 mm
   - Population standard deviation (\(\sigma\)) = 66 mm
   - We are interested in finding the rainfall corresponding to the top 15%, which corresponds to the 85th percentile.

2. **Find the Z-score for the 85th percentile:**
   The Z-score that corresponds to the cumulative probability of 0.85 can be found using the standard normal distribution.

3. **Calculate the corresponding rainfall value:**
   Use the Z-score formula to find the rainfall value:
   \[
   X = \mu + Z \times \sigma
   \]

Let's calculate this.The Z-score corresponding to the 85th percentile is approximately \(1.04\). The amount of yearly rainfall in Sydney that would be in the top 15% is approximately **202.40 mm**.

Would you like more details on this calculation, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. What is the rainfall value that corresponds to the median (50th percentile)?
2. How much rainfall would be in the bottom 10%?
3. What is the probability that the yearly rainfall exceeds 250 mm?
4. If the mean yearly rainfall increases to 150 mm, how does the 85th percentile change?
5. How would the standard deviation affect the top 15% if it increased to 80 mm?

**Tip:** Percentiles are useful for understanding the distribution of data and identifying thresholds that correspond to certain probabilities.

Learn how to calculate the amount of yearly rainfall in Sydney, Australia, corresponding to the top 15% using normal distribution and percentiles. This problem involves understanding the properties of normal distribution, calculating Z-scores, and applying the Z-score formula to determine rainfall values. Suitable for advanced high school or college-level students.

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