a professor has 184 students in his college the scores on the midterm are normally distributed with a mean of 72.3 and SD of 8.9 how many student in the class can be expected to receive a score between 82 and 90 what is the answer

To find the number of students expected to receive a score between 82 and 90, we need to use the properties of the normal distribution.

1. **Calculate the z-scores** for the given scores (82 and 90). The z-score is calculated using the formula:

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

   where:
   - \( x \) is the score
   - \( \mu \) is the mean
   - \( \sigma \) is the standard deviation

2. **Find the probability** associated with these z-scores using standard normal distribution tables or a calculator.

3. **Calculate the number of students** within that z-score range by multiplying the probability by the total number of students.

Let's go through the calculations step-by-step:

### Step 1: Calculate the z-scores

For \( x = 82 \):
\[
z_{82} = \frac{82 - 72.3}{8.9} = \frac{9.7}{8.9} \approx 1.09
\]

For \( x = 90 \):
\[
z_{90} = \frac{90 - 72.3}{8.9} = \frac{17.7}{8.9} \approx 1.99
\]

### Step 2: Find the probabilities

Using the standard normal distribution table or a calculator:

- The probability of \( z \leq 1.09 \) is approximately 0.8621
- The probability of \( z \leq 1.99 \) is approximately 0.9767

To find the probability of a score being between 82 and 90:
\[
P(82 \leq x \leq 90) = P(z \leq 1.99) - P(z \leq 1.09) = 0.9767 - 0.8621 = 0.1146
\]

### Step 3: Calculate the number of students

Multiply this probability by the total number of students:
\[
0.1146 \times 184 \approx 21.08
\]

Since the number of students must be an integer, we can round this to 21 students.

### Conclusion

Approximately 21 students can be expected to receive a score between 82 and 90.

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Do you want more details or have any questions?

Here are some related questions:

1. What is the probability that a student scores below 60?
2. How many students are expected to score above 85?
3. What is the probability that a student scores between 70 and 75?
4. How many students are expected to score within one standard deviation of the mean?
5. What is the z-score for a student who scores 95?
6. How many students are expected to score below 65?
7. What percentage of students score within two standard deviations of the mean?
8. How many students are expected to score exactly the mean score?

**Tip:** When working with normal distributions, always convert raw scores to z-scores to utilize standard normal distribution tables or calculators for finding probabilities.

Learn how to calculate the expected number of students with scores between 82 and 90 in a college class using the properties of normal distribution. Detailed step-by-step solution provided.

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