The average high School teacher annual salary is 43,000. Let teachers salary be normally distributed with a standard deviation of 18,000.  a) what percentage of high school teachers make between 40k and 50k? B) what percentage of high school teachers make more than 80k? 

Given the problem, we know that the teacher salaries are normally distributed with:

- Mean \( \mu = 43,000 \)
- Standard deviation \( \sigma = 18,000 \)

We can solve these types of problems using the **z-score** formula:

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

Where:
- \( x \) is the value of interest,
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation.

### Part (a): What percentage of high school teachers make between 40,000 and 50,000?

1. **For \( x_1 = 40,000 \):**

\[
z_1 = \frac{40,000 - 43,000}{18,000} = \frac{-3,000}{18,000} = -0.167
\]

2. **For \( x_2 = 50,000 \):**

\[
z_2 = \frac{50,000 - 43,000}{18,000} = \frac{7,000}{18,000} = 0.389
\]

Now, using the standard normal distribution (z-table), we can find the probabilities corresponding to these z-scores:

- For \( z_1 = -0.167 \), the cumulative probability is approximately 0.4338.
- For \( z_2 = 0.389 \), the cumulative probability is approximately 0.6517.

To find the percentage of teachers making between 40k and 50k, we subtract the cumulative probability of \( z_1 \) from \( z_2 \):

\[
P(40,000 \leq X \leq 50,000) = 0.6517 - 0.4338 = 0.2179
\]

Thus, **21.79%** of high school teachers make between 40k and 50k.

### Part (b): What percentage of high school teachers make more than 80,000?

For \( x = 80,000 \):

\[
z = \frac{80,000 - 43,000}{18,000} = \frac{37,000}{18,000} \approx 2.056
\]

Using the z-table, the cumulative probability for \( z = 2.056 \) is approximately 0.9802. This gives the proportion of teachers earning less than 80k.

To find the percentage of teachers making more than 80k, we subtract this value from 1:

\[
P(X > 80,000) = 1 - 0.9802 = 0.0198
\]

Thus, **1.98%** of high school teachers make more than 80k.

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Would you like further details or have any questions?

Here are some related questions for practice:

1. What percentage of teachers make less than 30k?
2. What percentage of teachers make between 35k and 45k?
3. What is the salary range for the middle 50% of teachers?
4. What percentage of teachers make less than the average salary?
5. How many teachers make between 60k and 70k if there are 1,000 teachers in total?

**Tip:** Understanding how z-scores relate to normal distribution tables is key to solving problems involving percentages in normal distributions.

The average high school teacher annual salary is 43,000. Let teachers' salaries be normally distributed with a standard deviation of 18,000. a) What percentage of high school teachers make between 40k and 50k? b) What percentage of high school teachers make more than 80k?

Solve the problem of calculating the percentage of high school teachers earning between 40k and 50k, and those earning more than 80k using normal distribution. This solution includes key concepts like z-scores and standard normal distribution, suitable for high school students in Grades 10-12.

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