Question

How fast can male college students run a mile? There’s lots of variation, of course. During World War II, physical training was required for male students in many colleges, as preparation for military service. That provided an opportunity to collect data on physical performance on a large scale. 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 7.01 minutes and standard deviation 0.7 minute.

It's good practice to draw a Normal curve on which this mean and standard deviation are correctly located. To do this, draw an unlabeled Normal curve, locate the points where the curvature changes (this is 1 standard deviation from the mean), then add number labels on the horizontal axis.

Use the Empirical Rule to answer the following questions.

1. What range of times covers the middle 99.7% of this distribution?
   
   From ______to _____minutes Round to 2 places.
   
2. What percent of these runners run the mile between 5.61 and 9.11 minutes?
   
   ______% Round to 2 places.
   
3. What percentage of the these running times are faster than 7.71 minutes?
   
   ________% Round to 2 places.

Answer 1

Let X be the time for the mile run by an able-bodied male student. X has normal distribution with mean

minutes and standard deviation minutes

The normal curve would look like this

The range of times covers the middle 99.7% of this distribution

Using the empirical rule we know that 99.7% of all the values in a normally distributed population lie within +-3 standard deviations from the mean.Since this is the middle range, the z score of lower range will be just the negative of the upper range.

Here the mean minutes and standard deviation minutes. +-3 standard deviation from mean would be

ans: The range of times covers the middle 99.7% of this distribution is From 4.91 to 9.11 minutes

Percent of these runners run the mile between 5.61 and 9.11 minutes is same as the probability that a randomly selected runner run the mile between 5.61 and 9.11 minutes

We need to find the distance of 5.61 and 9.11 from the mean in terms of number of standard deviations

z score of 5.61 is

5.61 is 2 standard deviations from the mean

9.11 is standard deviations from the mean.

We want the area under the normal curve from -2 standard deviation to +3 standard deviations from the mean.

We know the following for a normal curve from the empirical rule

99.7% of the values lie between +- 3 standard deviations.

That means half of 99.7% lie between the mean and +3 standard deviation from the mean.

That is 99.7/2=49.85% of the values lie between the mean and 3 standard deviations from the mean.

Next (from the empirical rule) we know that 95.5% of the values lie between +-2 standard deviations from the mean. That means half of this = 95.5/2=47.75% of values lie between the mean and -2 standard deviation from the mean.

We need the values between -2 and +3 standard deviations from the mean. That means 49.85+47.75=97.6% of the values lie between  -2 and +3 standard deviations from the mean

ans: percent of these runners run the mile between 5.61 and 9.11 minutes is 97.60%

percentage of the these running times are faster than 7.71 minutes is same as the probability that a randomly selected runner runs the mile in less than 7.71 minutes.
The z score of 7.71 is

That is we need the area under the normal curve to the left of +1 standard deviation from the mean.

We know from the empirical rule that 68% of the values in a normal distribution lie between +-1 standard deviation from the mean.

This means 68/2=34% of the values lie between the mean and +1 standard deviation from the mean.

We also know that 50% of the values are less than the mean.

That means 50+34=84% of the values are less than 1 standard deviation from the mean.

ans: percentage of the these running times are faster than 7.71 minutes is 84%