To calculate the probability that the annual precipitation in Tucson exceeds 12.14 inches using the cumulative distribution function (CDF) approach, we need to assume a probability distribution that models the data well. Since we calculated the mean (\(\mu\)) and standard deviation (\(\sigma\)), we can use a normal distribution as an approximation. Note that this method assumes the data is approximately normally distributed, which may not be perfectly accurate but provides a good estimate.

### Step 1: Parameters from Part (c)

- Mean (\(\mu\)): \(\approx 11.75\) inches
- Standard Deviation (\(\sigma\)): \(\approx 3.88\) inches

### Step 2: Convert the Problem to Standard Normal Distribution

To find the probability that the annual precipitation exceeds 12.14 inches, we need to convert the value into a z-score using the following formula:

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

Where:
- \(x = 12.14\) (the threshold value)
- \(\mu = 11.75\) (mean)
- \(\sigma = 3.88\) (standard deviation)

\[
z = \frac{12.14 - 11.75}{3.88} \approx \frac{0.39}{3.88} \approx 0.10
\]

### Step 3: Calculate Probability Using the Z-Score

The CDF of the standard normal distribution (denoted as \( \Phi(z) \)) gives the probability that a standard normal random variable is less than or equal to \(z\). To find the probability that the precipitation exceeds 12.14 inches, we need to calculate \( 1 - \Phi(z) \).

- \( P(X > 12.14) = 1 - \Phi(0.10) \)

Using standard normal distribution tables or a calculator:

\[
\Phi(0.10) \approx 0.5398
\]

\[
P(X > 12.14) = 1 - 0.5398 \approx 0.4602
\]

### Step 4: Probability Interpretation

The probability that the annual precipitation in Tucson exceeds 12.14 inches, based on the normal distribution approximation, is approximately **0.4602** or **46.02%**.

### Comparison with Frequency Diagram Calculation

- Using the actual data from the frequency diagram method, we found the probability to be 33%.
- Using the normal distribution and CDF method, we found the probability to be approximately 46.02%.

### Conclusion

The results vary slightly between the empirical frequency method and the CDF method. The CDF method, which assumes a normal distribution, gives a higher probability, indicating that the actual distribution may have some deviation from normality.

### Further Questions

1. How does the choice of normal distribution affect the accuracy of the probability calculations?
2. What are alternative methods to estimate the probability if the data is not normally distributed?
3. How sensitive are these probability calculations to changes in the mean and standard deviation?
4. What real-world factors could cause the precipitation data to deviate from a normal distribution?
5. How could the CDF method be applied to other environmental data sets?

### Tip

Always compare results from empirical methods (like frequency diagrams) with theoretical models (like CDFs) to understand the accuracy and appropriateness of each method for your data.

Learn how to calculate the probability that annual precipitation in Tucson exceeds 12.14 inches using the cumulative distribution function (CDF) approach. This problem involves applying statistical concepts like normal distribution approximation and the CDF of the standard normal distribution.

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