Solve probability, percentile, and more

Question

uppose that a category of world class runners are known to run a marathon (26 miles) in an average of 147 minutes with a standard deviation of 15 minutes. Consider 49 of the races.

Let X

 = the average of the 49 races.

a. Give the distribution of X.

 (Round your standard deviation to two decimal places.)

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b.Find the probability that the average of the sample will be between 146 and 149 minutes in these 49 marathons. (Round your answer to four decimal places.)

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

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d. Find the median of the average running times.

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Answer 1

According to the central limit theorem the sample mean of 49 races should be approximately normally distributed with a mean of 147 with a sd of 15/sqrt(49) = 15/7 . Now you can use Z scores and the Z table to solve the rest:

P(146<X<149) = P( (146-147)/(15/7) < Z < (149-147)/(15/7) ) go from here/. LMK if you need more help