Question

Average Salary of NFL Players The dataset NFLContracts2015 contains the yearly salary (in millions of dollars) from the contracts of all players on a National Football League (NFL) roster at the start of the 2015 season. \({ }^{19}\) (a) Use StatKey or other technology to select a random sample of 5 NFL contract YearlySalary values. Indicate which players you've selected and compute the sample mean. (b) Repeat part (a) by taking a second sample of 5 values, again indicating which players you selected and computing the sample mean. (c) Find the mean for the entire population of players. Include notation for this mean. Comment on the accuracy of using the sample means found in parts (a) and (b) to estimate the population mean.

Short answer

The process involves randomly selecting small samples (here, two samples of 5 players each), calculating their means, and comparing them with the population mean. These sample means provide estimates of the population mean. The accuracy depends on how close these sample means are to the population mean.

Step-by-step solution

Randomly Select 5 NFL Players

To conduct this task, use a dataset that contains the yearly salary values of all NFL players. It's important to ensure the selection process is random, to avoid any bias in the sample. A random selection method could be using random number generation in conjunction with a player list indexed by numbers.

Calculate Sample Mean for First 5 Players

After selecting 5 players randomly, calculate the sample mean for the first 5 players. The mean is calculated by adding up all the salary values and then dividing by the total number (5 in this case). Let's represent this mean as \(\mu_1\).

Repeat Random Selection and Calculation

The process of random selection and calculation of sample mean needs to be repeated to create another sample. Select 5 different players from the list and compute their mean salary. This is the second sample mean, \(\mu_2\).

Calculate Population Mean

Next, compute the mean for the entire population of players presented in the dataset. This is done by adding up all the players' salary values and dividing it by the total number of players. Let's denote this population mean as \(\mu_p\) .

Comparison of Sample Means and Population Mean

Finally, compare the sample means (\(\mu_1\) and \(\mu_2\)) with the population mean (\(\mu_p\)). This will provide insights into the accuracy of using sample means to estimate the population mean. If the sample means are close to the population mean, they're good estimates; if not, the accuracy may be in question.

Key concepts

Random Sampling

In any statistical analysis, the keystone for valid inferences is the way in which we sample the whole population of interest. Random sampling is a technique where every individual member of the population has an equal chance of being included in the sample. Think of it as a lottery where every NFL player's contract could be 'picked' independently of the others.

For the NFL salary study, a random sample of 5 players must be chosen from the entire list of players. This represents a tiny fraction of the population that we will examine to infer something about the whole group. It helps to prevent bias; for example, intentionally or unintentionally choosing only the top-paid players would skew the results. The lottery-like randomness ensures that the sample represents the broader NFL salary structure.

Using technology like StatKey or other random number generators paired with a numbered list of players simplifies this process. With the touch of a button, you can generate a truly random subset of players irrespective of their standing. This approach is fundamental in ensuring that your statistics—like sample mean—reflect genuine attributes of the population.

Sample Mean Calculation

Once we have a random sample, the next step is to determine the sample mean, which is a measure of the central tendency of the sample salaries. It's calculated by summing all the individual salaries within the sample and then dividing by the number of players in the sample.

To illustrate, let's say we picked 5 players randomly and their salaries are: \(1 million, \)3 million, \(0.8 million, \)2.2 million, and $4 million. The sample mean would be calculated as follows: \( \frac{1 + 3 + 0.8 + 2.2 + 4}{5} = \frac{11}{5} = 2.2 \) million dollars. This figure, while simple to compute, is packed with insights about the group we're examining. It's used as a point estimate of the average NFL player's salary.

The key in this computation is precision. Each selection and calculation must be performed with care to ensure that the resulting mean is as accurate as possible. Multiple sample means, like those calculated from different random samples, give a fuller picture by providing different snapshots of the same population.

Population Mean Estimation

The population mean estimation is the process of using sample data to deduce the average salary of the entire roster of NFL players. It's what statisticians aim to approximate when they can't examine every single player due to practical limitations.

In our example,5 each sample mean gives us a separate estimation of the true population mean. When comparing these estimations (from the first and second samples) to the actual population mean, we gain insight into their accuracy. If they're fairly close to the true mean, we can say our sampling method does a decent job of estimating. Deviations indicate potential flaws or simply the natural sampling variability.

Computing the true mean salary involves adding up all players' salaries (\( \text{let's say it's } S \text{ dollars for the entire list}\) and dividing by the total number of players (\( N \text{, for instance}\)\( \frac{S}{N} = \mu_p \) where \( \mu_p \) denotes the population mean. This value is the benchmark against which we measure our sample mean results, identifying variances and trends, and ultimately, judging the effectiveness of our samples as representatives of the larger group.