Question

3.34 A Sampling Distribution for Average Salary of NFL Players Use StatKey or other technology to generate a sampling distribution of sample means using a sample of size \(n=5\) from the YearlySalary values in the dataset NFLContracts2015, which gives the total and yearly money values from the contracts of all NFL players in 2015 . (a) What are the smallest and largest YearlySalary values in the population? (b) What are the smallest and largest sample means in the sampling distribution? (c) What is the standard error (that is, the standard deviation of the sample means) for the sampling distribution for samples of size \(n=5 ?\) (d) Generate a new sampling distribution with samples of size \(n=50 .\) What is the standard error for this sampling distribution?

Short answer

Without the actual dataset it's impossible to provide exact numeric answers. However, to answer conceptually: (a) The smallest and largest 'YearlySalary' can be found by sorting the dataset. (b) The smallest and largest sample means are identified from the sampling distribution generated from the dataset. (c) The standard error for \(n = 5\) is calculated by diving the standard deviation by \(\sqrt{5}\). (d) The standard error for \(n = 50\) is calculated by diving the standard deviation by \(\sqrt{50}\), and should be smaller than for \(n = 5\), demonstrating that larger samples yield more precise estimates. Exact figures require precise computations from the dataset.

Step-by-step solution

Understand the dataset

Recognize that 'YearlySalary' values in NFLContracts2015 dataset refers to salary of each NFL player. Therefore, each player represents a unit of population.

Identify Smallest and Largest Salaries

To find the smallest and largest values in the population, organize the 'YearlySalary' values of the dataset from smallest to largest.

Generate a sampling distribution

Use a data analysis software such as StatKey to generate a sampling distribution of sample means using a sample size of \(n = 5\) from the dataset.

Identify smallest and largest sample means

Once the sampling distribution is generated, identify the smallest and largest 'sample means' in the sampling distribution. Note that the 'sample mean' is the mean of a single random sample from the population.

Calculate Standard Error for \(n = 5\)

The standard error is the standard deviation of the sampling distribution. It can be calculated using the formula: \(SE = \sigma / \sqrt{n}\) where \(\sigma\) is the standard deviation of the population, and \(n\) is the sample size. In this case, \(n = 5\). If the standard deviation of the population is unknown, it can be approximated using the standard deviation of the sample.

Generate a new sampling distribution for \(n = 50\)

Generate a new sampling distribution with samples of size \(n = 50\). This will show how the increase in sample size affects the shape of the sampling distribution and the standard error.

Calculate Standard Error for \(n = 50\)

Calculate the standard error again, using \(n = 50\) in the formula from Step 5. This should result in a smaller standard error, demonstrating that increasing the sample size increases the precision of the sampling distribution.