Question

24. Running a mile A study of $12,000$able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal 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. Find $P(Y<6)$ and interpret the result. Follow the
four-step process.

Short answer

The probability is $0.0668$. And there is $6.68\%$chances that student's run would be less than $6$ minutes.

Step-by-step solution

Given information

Normal with mean $7.11$minutes and standard deviation $0.74$ minute.

Explanation

Let's say $X$is a random variable that has a normal distribution with a mean of $7.11$and a standard deviation of $0.74$.

The$z-$score is calculated by dividing the value reduced by the mean by the standard deviation:

$z=\frac{x-\mu }{\sigma }$
Where, $x$indicates raw score, $\mu$indicates population mean, and $\sigma$indicates population standard deviation.

$P(Y<6)$can be calculated as:

$$\begin{gathered} P(Y<6)=P\left(\frac{y-\mu }{\sigma }<\frac{9-\mu }{\sigma }\right) \\ =P\left(Z<\frac{6-7.11}{0.74}\right) \\ =P(Z<-1.50) \end{gathered}$$