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Answer :
Sure! Let's break down the solution step-by-step for each part of the question.
Part a)
To find the range of ages where approximately 99% of all NFL players' retirement ages fall, we can use the properties of the normal distribution.
In a normal distribution:
- Approximately 68% of the data lies within 1 standard deviation (σ) of the mean (μ).
- Approximately 95% lies within 2 standard deviations.
- Approximately 99% lies within 3 standard deviations.
Given:
- Mean (μ) = 33.7 years
- Standard deviation (σ) = 2.1 years
We calculate the values within 3 standard deviations of the mean:
- Lower value = μ - 3σ = 33.7 - 3(2.1) = 33.7 - 6.3 = 27.4 years
- Upper value = μ + 3σ = 33.7 + 3(2.1) = 33.7 + 6.3 = 40.0 years
So, approximately 99% of all NFL players' retirement ages fall between 27.4 years and 40.0 years.
Part b)
To estimate the percentage of NFL players who retire between the ages of 29.5 and 37.9 years, we need to convert these ages to z-scores and then use the cumulative distribution function (CDF) for a normal distribution.
Given:
- Lower bound = 29.5 years
- Upper bound = 37.9 years
Calculate the z-scores:
- [tex]z_{\text{lower}} = \frac{29.5 - 33.7}{2.1} = \frac{-4.2}{2.1} = -2.0[/tex]
- [tex]z_{\text{upper}} = \frac{37.9 - 33.7}{2.1} = \frac{4.2}{2.1} = 2.0[/tex]
Using the properties of the normal distribution, the cumulative distribution function (CDF) tells us the probability that a normally distributed random variable will be less than or equal to a given value.
From statistical tables or using a calculator:
- [tex]\text{CDF}(z_{\text{lower}} = -2.0) \approx 0.0228[/tex]
- [tex]\text{CDF}(z_{\text{upper}} = 2.0) \approx 0.9772[/tex]
The percentage of players retiring between 29.5 and 37.9 years is:
- [tex]\text{CDF}(2.0) - \text{CDF}(-2.0) = 0.9772 - 0.0228 = 0.9544[/tex]
So, approximately 95.45% of NFL players retire between the ages of 29.5 and 37.9 years.
Part c)
To estimate the percentage of NFL players who retire between the ages of 31.6 and 35.8 years, we follow the same process of converting the ages to z-scores.
Given:
- Lower bound = 31.6 years
- Upper bound = 35.8 years
Calculate the z-scores:
- [tex]z_{\text{lower}} = \frac{31.6 - 33.7}{2.1} = \frac{-2.1}{2.1} = -1.0[/tex]
- [tex]z_{\text{upper}} = \frac{35.8 - 33.7}{2.1} = \frac{2.1}{2.1} = 1.0[/tex]
Using the properties of the normal distribution:
- [tex]\text{CDF}(z_{\text{lower}} = -1.0) \approx 0.1587[/tex]
- [tex]\text{CDF}(z_{\text{upper}} = 1.0) \approx 0.8413[/tex]
The percentage of players retiring between 31.6 and 35.8 years is:
- [tex]\text{CDF}(1.0) - \text{CDF}(-1.0) = 0.8413 - 0.1587 = 0.6826[/tex]
So, approximately 68.27% of NFL players retire between the ages of 31.6 and 35.8 years.
Summary
To summarize:
a) Approximately 99% of all NFL players' retirement ages fall between 27.4 years and 40.0 years.
b) Approximately 95.45% of NFL players retire between the ages of 29.5 and 37.9 years.
c) Approximately 68.27% of NFL players retire between the ages of 31.6 and 35.8 years.
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