A population is normally distributed with mean 36.8 and standard deviation 2.5. Find the following probability. (Round your answer to four decimal places.)
p( 36.8 < x < 39.3 )

To find the probability \( P(36.8 < X < 39.3) \), where \( X \) is normally distributed with a mean \( \mu = 36.8 \) and standard deviation \( \sigma = 2.5 \), we need to standardize the values using the Z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

### Step 1: Find the Z-scores
1. For \( X = 36.8 \):
   \[
   Z_1 = \frac{36.8 - 36.8}{2.5} = 0
   \]
2. For \( X = 39.3 \):
   \[
   Z_2 = \frac{39.3 - 36.8}{2.5} = \frac{2.5}{2.5} = 1
   \]

### Step 2: Use the standard normal distribution table
- The probability corresponding to \( Z_1 = 0 \) is 0.5000.
- The probability corresponding to \( Z_2 = 1 \) is 0.8413.

### Step 3: Calculate the probability
The probability \( P(36.8 < X < 39.3) \) is the difference between the two probabilities:

\[
P(36.8 < X < 39.3) = P(Z_2) - P(Z_1) = 0.8413 - 0.5000 = 0.3413
\]

### Final Answer:
\[
P(36.8 < X < 39.3) = 0.3413
\]

Would you like further explanation or details?

Here are 5 related questions to expand on this:
1. How do you find probabilities for other intervals using Z-scores?
2. What is the meaning of a Z-score in statistics?
3. How does changing the standard deviation affect the probability in a normal distribution?
4. What if the population were not normally distributed—how would you approach this problem?
5. How do you calculate probabilities for values outside one standard deviation from the mean?

**Tip:** Always ensure to standardize values with the Z-score formula when dealing with normal distributions.

A population is normally distributed with mean 36.8 and standard deviation 2.5. Find the following probability. (Round your answer to four decimal places.) p( 36.8 < x < 39.3 )

Learn how to calculate the probability of P(36.8 < X < 39.3) for a normally distributed population with a mean of 36.8 and a standard deviation of 2.5. This solution demonstrates step-by-step Z-score calculations and uses the standard normal distribution table. Ideal for students in Grades 10-12.

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