Question

Marathon run times follow a normal distribution with mean equal to four hours and standard deviation equal to 2* X minutes where X is the position of the first letter of your last name in the alphabet. For example, if your last name starts with C then the standard deviation is 2*3 = 6 minutes because C is the third letter in the alphabet.

A) What are the marathon times for the middle 95% of the runners?
B) How fast did the top 16% of fastest marathon runners go?
C) Your marathon time was 3 hours and 50 minutes. What percentage of runners went faster than you?
D) Find how fast the fastest 5% of runners went.

Answer 1

The marathon times for the middle 95% of runners can be found by using z-scores. The top 16% of fastest marathon runners go faster than a certain time. The percentage of runners that went faster than a specific marathon time can be determined by calculating the z-score and looking it up in a standard normal distribution table. The fastest 5% of runners can be found by using z-scores as well.

Step-by-step explanation:

To find the marathon times for the middle 95% of the runners, we need to find the interval that contains 95% of the data. Since marathon run times follow a normal distribution, we can use the z-score formula to find the interval. The z-score corresponding to 95% is 1.96. The lower bound of the interval is the mean minus 1.96 times the standard deviation, and the upper bound is the mean plus 1.96 times the standard deviation. So, the marathon times for the middle 95% of the runners are between 4 - 1.96(2X) hours and 4 + 1.96(2X) hours.

To find how fast the top 16% of fastest marathon runners go, we need to find the z-score corresponding to the top 16%. The z-score corresponding to the top 16% is 0.95. The upper bound of the interval is the mean plus 0.95 times the standard deviation. So, the top 16% of fastest marathon runners go faster than 4 + 0.95(2X) hours.

To find the percentage of runners that went faster than a marathon time of 3 hours and 50 minutes, we need to find the z-score corresponding to that time. The z-score is calculated by subtracting the mean from the marathon time and dividing by the standard deviation. Then, we can look up the percentage corresponding to the z-score in a standard normal distribution table. The percentage of runners that went faster than 3 hours and 50 minutes is the complement of the percentage corresponding to the z-score.

To find how fast the fastest 5% of runners went, we need to find the z-score corresponding to the top 5%. The z-score corresponding to the top 5% is 1.645. The upper bound of the interval is the mean plus 1.645 times the standard deviation. So, the fastest 5% of runners went faster than 4 + 1.645(2X) hours.