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)
Responses

55
55

950
950

945
945

445

Answer 1

To determine how many students are shorter than 121 centimeters, we can use the properties of the normal distribution.

1. **Calculate the z-score** for 121 centimeters using the formula:

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

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

Plugging in the values:

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

2. **Look up the z-score** in the standard normal distribution table (or use a calculator) to find the proportion of students that are shorter than 121 cm.

A z-score of 1.6 corresponds to a cumulative probability of approximately 0.9452 (or 94.52%) of the distribution.

3. **Calculate the number of students shorter than 121 cm**:

\[
\text{Number of students} = \text{Total students} \times \text{Cumulative probability}
\]

\[
\text{Number of students} = 1000 \times 0.9452 \approx 945.2
\]

We round this to approximately 945 students.

Therefore, the number of students shorter than 121 centimeters is **945**.

The correct response is: **945**.