Short Answer 
Approximately 99% of NFL players retire between ages 27.4 and 40.0 years, while around 95.45% retire between 29.5 and 37.9 years. Additionally, about 68.27% retire between ages 31.6 and 35.8 years.
Step 1: Determine the Retirement Age Range
To find the age range where approximately 99% of NFL players retire, we utilize the properties of the normal distribution. Given a mean (Œº) of 33.7 years and a standard deviation (œE) of 2.1 years, we calculate:
- Lower value: Œº Р3œE = 33.7 Р3(2.1) = 27.4 years
- Upper value: Œº + 3œE = 33.7 + 3(2.1) = 40.0 years
Thus, approximately 99% of NFL players’ retirement ages fall between 27.4 and 40.0 years.
Step 2: Calculate Retirement Age Probability (29.5 to 37.9 years)
Next, to find the percentage of players retiring between 29.5 and 37.9 years, we must convert these ages into z-scores. The calculations yield:
- Lower z-score: zlower = (29.5 – 33.7) / 2.1 = -2.0
- Upper z-score: zupper = (37.9 – 33.7) / 2.1 = 2.0
Using the cumulative distribution function (CDF), we find that approximately 95.45% of NFL players retire within this age bracket.
Step 3: Estimate Retirement Age Probability (31.6 to 35.8 years)
Finally, for the age range of 31.6 to 35.8 years, we again convert these ages to z-scores:
- Lower z-score: zlower = (31.6 – 33.7) / 2.1 = -1.0
- Upper z-score: zupper = (35.8 – 33.7) / 2.1 = 1.0
Applying the CDF, we determine that approximately 68.27% of NFL players retire within this specified age range.