Question

In a school with 1,000 students, the heights of students are normally distributed with a mean of 113 centimeters and a standard deviation of 5 centimeters. How many students are shorter than 121 centimeters? (1 point) 945 55 950 445

Answer 1

To find out how many students are shorter than 121 centimeters in a normally distributed population, we will first calculate the z-score for 121 centimeters using the formula:

\[
z = \frac{(X - \mu)}{\sigma}
\]

where:
- \(X\) is the value we are interested in (121 cm),
- \(\mu\) is the mean (113 cm),
- \(\sigma\) is the standard deviation (5 cm).

Substituting the values into the formula:

\[
z = \frac{(121 - 113)}{5} = \frac{8}{5} = 1.6
\]

Next, we need to find the cumulative probability for a z-score of 1.6 using the standard normal distribution table or a calculator. The cumulative probability \(P(Z < 1.6)\) is approximately 0.9452.

This means that about 94.52% of students are shorter than 121 centimeters.

Now we will calculate how many students this percentage corresponds to in a school with 1,000 students:

\[
\text{Number of students shorter than 121 cm} = 0.9452 \times 1000 \approx 945
\]

So the number of students shorter than 121 centimeters is approximately 945.

Thus, the answer is **945**.