In an all boys school, the heights of the student body are normally distributed with a mean of 68 inches and a standard deviation of 2.5 inches. What percentage of the students are between 65 and 72 inches tall, to the nearest tenth?
To find the percentage of students between 65 and 72 inches tall, we first need to calculate the z-scores for each height.
For 65 inches:
\[ z = \frac{65 - 68}{2.5} = -1.2 \]
For 72 inches:
\[ z = \frac{72 - 68}{2.5} = 1.6 \]
Using a standard normal distribution table or calculator, we find the corresponding probabilities for these z-scores:
- For a z-score of -1.2, the probability is 0.1151
- For a z-score of 1.6, the probability is 0.9452
To find the percentage of students between 65 and 72 inches, we subtract the probability corresponding to 65 inches from the probability corresponding to 72 inches:
\[ 0.9452 - 0.1151 = 0.8301 \]
So approximately 83.0% of the students are between 65 and 72 inches tall.