In an all boys school, the heights of the student body are normally distributed with a mean of 67 inches and a standard deviation of 3.5 inches. Out of the 793 boys who go to that school, how many would be expected to be taller than 60 inches tall, to the nearest whole number?

To solve this problem, we need to find the probability that a student is taller than 60 inches using the normal distribution, and then apply this probability to the total number of students (793) to find the expected number.

### Step 1: Convert the height to a z-score
The z-score is calculated using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \( X \) is the value of interest (60 inches),
- \( \mu \) is the mean (67 inches),
- \( \sigma \) is the standard deviation (3.5 inches).

Substituting the values:
\[
z = \frac{60 - 67}{3.5} = \frac{-7}{3.5} = -2
\]

### Step 2: Find the probability corresponding to the z-score
We need to find the probability that a student is taller than 60 inches, which is \( P(X > 60) \). This is equivalent to \( P(Z > -2) \).

Using the standard normal distribution table:
\[
P(Z > -2) = 1 - P(Z < -2) = 1 - 0.0228 = 0.9772
\]

### Step 3: Calculate the expected number of students taller than 60 inches
Multiply the probability by the total number of students:
\[
\text{Expected number} = 0.9772 \times 793 \approx 774.92
\]

Rounding to the nearest whole number:
\[
\text{Expected number} \approx 775
\]

### Final Answer:
Approximately **775** students would be expected to be taller than 60 inches.

Would you like any further details or have any questions? Here are 5 related questions you might find interesting:

1. What is the probability that a student is shorter than 64 inches?
2. How many students would be expected to be between 64 and 70 inches tall?
3. What height corresponds to the top 10% of the students?
4. How would the result change if the standard deviation was 4 inches instead of 3.5 inches?
5. What proportion of students are expected to be within one standard deviation of the mean?

**Tip:** When dealing with normal distributions, converting to z-scores simplifies the process of finding probabilities using standard normal distribution tables.

Calculate how many students are expected to be taller than 60 inches in an all boys school with a mean height of 67 inches and a standard deviation of 3.5 inches. This problem explores key statistical concepts, including normal distribution, z-scores, and probability. Suitable for Grades 10-12.

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