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To solve this problem, let’s go through the steps for calculating the probability that a teacher’s salary falls between $36,000 and $51,000, given a normal distribution:

### Given Data:
- **Mean (μ)** = $47,750
- **Standard Deviation (σ)** = $5,680
- We want to find $ P(36,000 \leq X \leq 51,000) $.

### Steps:
1. **Convert the values $36,000 and $51,000 into Z-scores** to standardize them.
   The Z-score formula is:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]

2. **Calculate the Z-score for $36,000**:
   \[
   Z_{36,000} = \frac{36,000 - 47,750}{5,680} \approx -2.0655
   \]

3. **Calculate the Z-score for $51,000**:
   \[
   Z_{51,000} = \frac{51,000 - 47,750}{5,680} \approx 0.5722
   \]

4. **Find the probabilities** for these Z-scores:
   - Using a Z-table or calculator:
     - $ P(Z \leq -2.0655) \approx 0.0195 $
     - $ P(Z \leq 0.5722) \approx 0.7163 $

5. **Calculate the probability** between these two Z-scores:
   \[
   P(36,000 \leq X \leq 51,000) = P(Z \leq 0.5722) - P(Z \leq -2.0655)
   \]
   \[
   P(36,000 \leq X \leq 51,000) \approx 0.7163 - 0.0195 = 0.6968
   \]

### Answer:
The probability that a randomly selected teacher earns between $36,000 and $51,000 a year is approximately **0.6968**, or **69.68%**.

Would you like further details or explanations on any of these steps?

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Here are five related questions to expand your understanding:

1. How would you calculate the probability for a range above or below a given salary, instead of between two values?
2. What is the probability that a teacher earns more than $60,000 a year given the same mean and standard deviation?
3. How would a higher or lower standard deviation affect the probability calculation in this scenario?
4. Can you explain why Z-scores are useful when working with normal distributions?
5. How could you determine the salary range that covers the middle 50% of teachers’ salaries?

**Tip**: Remember that for normal distributions, probabilities can be found using Z-tables, but calculators with statistical functions can simplify this process significantly.

A randomly selected teacher earns between $36,000 and $51,000 a year. Given the average annual salary for all U.S. teachers is $47,750 with a standard deviation of $5,680, calculate the probability that a teacher's salary falls within this range using a normal distribution.

Learn how to calculate the probability that a teacher's salary falls between $36,000 and $51,000, given a normal distribution with a mean of $47,750 and standard deviation of $5,680. This solution involves finding Z-scores and using a normal distribution table, suitable for college-level statistics students.

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