QQuestionStatistics
QuestionStatistics
The retirement age for NFL players is bell-shaped (normally distributed) with:
A mean (μ) of 33.7 years old,
A standard deviation (σ) of 2.1 years.
a) What two values will approximately 99% of all NFL players' retirement ages fall between?
b) Estimate the percentage of NFL players who retire between the ages of 29.5 and 37.9 years.
c) Estimate the percentage of NFL players who retire between the ages of 31.6 and 35.8 years.
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Answer
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Step 1:I'll solve this problem step by step using the standard normal distribution (z-score) method:
Step 2:: Understand the Normal Distribution Parameters
- Mean (μ) = 33.7 years - Standard Deviation (σ) = 2.1 years
Step 3:: Part (a) - Finding 99% Interval
- Upper bound: $$33.7 + (2.576 \times 2.1) = 33.7 + 5.41 = 39.11
- For 99% coverage, we need ±2.576 standard deviations from the mean Calculations:
Step 4:: Part (b) - Calculating Z-Scores
- Lower bound z-score: $$z_{1} = \frac{29.5 - 33.7}{2.1} = -2.00
- Upper bound z-score: z_{2} = \frac{37.9 - 33.7}{2.1} = 2.00
Step 5:: Finding Probability Using Standard Normal Distribution
- From z-table, area between - 2.00 and 2.00 is approximately 0.9544 or 95.44%
Step 6:: Part (c) - Calculating Z-Scores
- Lower bound z-score: $$z_{1} = \frac{31.6 - 33.7}{2.1} = -1.00
- Upper bound z-score: z_{2} = \frac{35.8 - 33.7}{2.1} = 1.00
Step 7:: Finding Probability
- From z-table, area between - 1.00 and 1.00 is approximately 0.6827 or 68.27%
Final Answer
a) NFL players' retirement ages will fall between 28.29 and 39.11 years (99% interval) b) Approximately 95.44% of NFL players retire between 29.5 and 37.9 years c) Approximately 68.27% of NFL players retire between 31.6 and 35.8 years
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