Question

Suppose that a category of world class runners are known to run a marathon (26 miles) in an average of 146 minutes with a standard deviation of 14 minutes. Consider 49 of the races.

Let X = the average of the 49 races.

I. Give the distribution of X-bar. (Round your standard deviation to two decimal places.)

X-bar ~ ____ ( ____ , ____ ).

II. Find the probability that the runner will average between 143 and 148 minutes in these 49 marathons. (Round your answer to four decimal places.)

III. Find the 60th percentile for the average of these 49 marathons. (Round your answer to two decimal places.) _____ min.

IV. Find the median of the average running times.

_____ min.

Is this a Central Limit Theorem or a normal distribution?

Answer 1

Solution: I. Give the distribution of X-bar. (Round your standard deviation to two decimal places.)

Answer:

II. Find the probability that the runner will average between 143 and 148 minutes in these 49 marathons. (Round your answer to four decimal places.)

Answer: We have to find

When , then we have:

Similarly when , then we have:

Therefore we have to find

Now using the standard normal table we have:

  

Hence the probability that the runner will average between 143 and 148 minutes in these 49 marathons is 0.7745

III. Find the 60th percentile for the average of these 49 marathons.

Answer: minutes.

IV. Find the median of the average running times.

Answer: minutes

Is this a Central Limit Theorem or a normal distribution?

Answer: This is a Central Limit Theorem.