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Expert: - 4/5/2009


Question
I have been given the first runner to finish took 54 mins and 2000 runners took part and given table shows the cumulative numbers of runners who had finished by various times. In the table,the column with title time after start (hours) are 1,1.5,2,2.5,3 and 4.The question are by how much should your answer for the mean be modified,given the additional information that the last runner to finish took exactly 3.5 hours? I dont know what formula to use?Can you tell me the steps how to do this?I will calculate it myself. Thanks a million

Answer
We are going to approximate with a normal distribution.
Assume that there will be 1,000 above the average and 1,000 below the average.

Using a normal distribution, find 0.001.  This will determine how many standard deviations are between 1st place and the average.
Note that the average is 210 minutes.  The difference between there and the first runner = 201-54 minute, or 147 minutes.

Here is a good table to use on the normal distribution:
http://www.stat.ucla.edu/~ywu/teaching/normal.pdf

Divide this by the amount found to be 0.001 from the left side.
NOte that the normal table is reflective, so on this normal table,
find 1-0.001 = 0.999.  This looks to be about 3.1 standard deviations from the middle.  To find a standard deviation, divide 147 (which we found to be the difference from the man) by 3.1.

Multiply the hours (1. 1.5, 2, 2.5, 3, and 4) by 60 to get minutes.
Divide each of these by the standard deviation to get what value to look up on the table sent.

To look up -2.86, for example, go to the row that says -2.8 on the left side, then acrose to the column labeled .06.  That's the value to use: 0.0021.  I'm sure there will be different values to look up, but that shows how to do it.

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