To determine how many students are shorter than 121 centimeters, we can use the properties of the normal distribution.
-
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:
\[ z = \frac{121 - 113}{5} = \frac{8}{5} = 1.6 \]
-
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.
-
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.