Question

a study of 12,000 able - bodied male students at the university of illinois found that their times for the mile run are approximately normally distributed, with mean 7.11 minutes and standard deviation 0.74 minute. choose a student at random from this group and call his time for the mile ( y ).( p(y < 6)=) (round to 4 decimal places. leave your answer in decimal form.) describe this probability in words. there is about a probability that a from this group will run the mile in less than 6 minutes.

Explanation:

Step1: Identify the distribution and parameters

We have a normal distribution \( Y \sim N(\mu = 7.11, \sigma = 0.74) \). We need to find \( P(Y < 6) \).

Step2: Calculate the z-score

The z-score formula is \( z=\frac{x - \mu}{\sigma} \). Substituting \( x = 6 \), \( \mu = 7.11 \), and \( \sigma = 0.74 \), we get:

$$ z=\frac{6 - 7.11}{0.74}=\frac{- 1.11}{0.74}\approx - 1.50 $$

Step3: Find the probability from z-table

We need to find \( P(Z < - 1.50) \) (where \( Z \) is the standard normal variable). Looking at the standard normal table, the area to the left of \( z=-1.50 \) is approximately \( 0.0668 \).

Answer:

\( 0.0668 \)