The annual rainfall in a certain region is approximately normally distributed with mean 41.3 inches and standard deviation 6 inches. Round answers to the nearest tenth of a percent.

a) What percentage of years will have an annual rainfall of less than 43 inches?
%
b) What percentage of years will have an annual rainfall of more than 39 inches?
%
c) What percentage of years will have an annual rainfall of between 37 inches and 42 inches?
%

To solve these problems, we'll use the properties of the normal distribution. The normal distribution is defined by two parameters: the mean (μ) and the standard deviation (σ). Given the problem:

- Mean (μ) = 41.3 inches
- Standard deviation (σ) = 6 inches

We can find the percentage of years by calculating the z-scores and then using the standard normal distribution table (or a calculator).

### (a) Percentage of years with rainfall less than 43 inches
To find this, we first calculate the z-score for 43 inches.

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

Where \(x = 43\).

\[
z = \frac{43 - 41.3}{6} \approx 0.2833
\]

Now, we find the cumulative probability for \(z = 0.2833\).

### (b) Percentage of years with rainfall more than 39 inches
Similarly, calculate the z-score for 39 inches.

\[
z = \frac{39 - 41.3}{6} \approx -0.3833
\]

The percentage of years with more than 39 inches of rainfall is the complement of the cumulative probability up to this z-score.

### (c) Percentage of years with rainfall between 37 inches and 42 inches
Calculate the z-scores for 37 inches and 42 inches:

\[
z_{37} = \frac{37 - 41.3}{6} \approx -0.7167
\]

\[
z_{42} = \frac{42 - 41.3}{6} \approx 0.1167
\]

We find the cumulative probabilities for these z-scores and then find the difference between them.

Let's calculate the corresponding percentages using the z-scores.Here are the results:

a) The percentage of years with an annual rainfall of less than 43 inches is **61.2%**.

b) The percentage of years with an annual rainfall of more than 39 inches is **64.9%**.

c) The percentage of years with an annual rainfall of between 37 inches and 42 inches is **31.0%**.

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to explore further:

1. What is the probability of having exactly the mean annual rainfall of 41.3 inches?
2. How would the percentages change if the standard deviation were smaller?
3. What percentage of years will have an annual rainfall of less than 35 inches?
4. What is the z-score for an annual rainfall of 50 inches?
5. How does the shape of the normal distribution curve change with varying standard deviations?
6. What is the probability of having an annual rainfall greater than the mean plus one standard deviation?
7. What percentage of years will have an annual rainfall between 40 inches and 45 inches?
8. How does the area under the normal distribution curve relate to probability?

**Tip:** When working with normal distributions, always check if the problem involves standardizing the variable using the z-score formula to easily find probabilities and percentages.

Explore the normal distribution and its application to annual rainfall statistics. Calculate probabilities for rainfall amounts using mean 41.3 inches and standard deviation 6 inches. Learn how to determine percentages of years with specific rainfall thresholds.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation