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The times of the finishers in the New York City 10km run are normally distributed with...

The times of the finishers in the New York City 10km run are normally distributed with a mean of µ minutes and standard deviation of 9 minutes. It is known that 80% of finishers have a finish time greater than 53.44 minutes. Let X denote the finishing time for finishers in this race.

  1. Find the mean finishing time (µ). Note: Provide the R code and output for the z-value or finding the area under the standard normal curve.
  2. In 2013 approximately 7748 individuals took part in the run. A random sample of 20 individuals is drawn and their finishing times are recorded. Assuming everyone finished the run, what is the probability that the average finishing time of the 20 selected individuals is greater than 55 minutes? Note: Mention the R code and output for the z-value or finding the area under the standard normal curve.
  3. A second, independent sample of individuals is drawn from this population. How large of a sample must be drawn if the probability that the average finishing time less than 59 is 30%?  Note: Show the R code and output for the z-value or for finding the area under the standard normal curve.
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