First, let's assume your last name starts with the letter 'B'. 'B' is the second letter in the alphabet, which we would denote as 2.
Since the normal distribution for the marathon times has a mean of four hours or 240 minutes, and the standard deviation is calculated by multiplying the position of the first letter of your last name in the alphabet by 2, we then multiply 2 (the position of 'B') by 2, giving us a standard deviation of 4 minutes.
To find the average 95% of runners' marathon times, we use the statistical rule that about 95% of values in a normal distribution falls between +/- 2 standard deviations from the mean. This means we subtract and add 2 times the standard deviation from the mean. Hence, the lower boundary of the 95% interval would be 240 - 2 * 4 = 232 minutes and the upper boundary would be 240 + 2 * 4 = 248 minutes. Therefore, 95% of runners finish the marathon between 232 and 248 minutes.
To find out how fast the top 16% of marathon runners finished, we use the inverse of the cumulative distribution function, also known as the percent point function. Using this function, it is calculated that the fastest 16% of runners finish in approximately 243.98 minutes.
Finally, to find out what time the slowest 2.5% of athletes finished, we again use the cumulative distribution function's inverse, which is also known as the percent point function. From this, we find that the slowest 2.5% of athletes complete the marathon in approximately 247.84 minutes.
In summary, the times for the runners are as follows:
- The average 95% of runners finish between 232 and 248 minutes.
- The fastest 16% of runners finish in approximately 243.98 minutes.
- The slowest 2.5% of athletes finish around 247.84 minutes.