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Chapter 3: Problem 6

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, \((b)\) median, (c) mode, (d) midrange, and then answer the given question. Listed below are the weights in pounds of 11 players randomly selected from the roster of the Seattle Seahawks when they won Super Bowl XLVIII (the same players from the preceding exercise). Are the results likely to be representative of all National Football League (NFL) players? $$ \begin{array}{lllllllllll} 189 & 254 & 235 & 225 & 190 & 305 & 195 & 202 & 190 & 252 & 305 \end{array} $$

Short Answer

Expert verified
Mean: 249.27 lbs, Median: 225 lbs, Mode: 190 and 305 lbs, Midrange: 247 lbs. The sample may not represent all NFL players due to small size and team bias.

Step by step solution

01

Arrange data in ascending order

First, arrange the given weights in ascending order: \[189, 190, 190, 195, 202, 225, 235, 252, 254, 305, 305.\]
02

Calculate the Mean

To find the mean, add all the weights together and divide by the number of players. \[ \text{Mean} = \frac{189 + 190 + 190 + 195 + 202 + 225 + 235 + 252 + 254 + 305 + 305}{11} \]\[ \text{Mean} = \frac{2742}{11} \]\[ \text{Mean} \thickapprox 249.27 \text{ pounds}\]
03

Determine the Median

The median is the middle number in a sorted list. Since we have 11 players, the median will be the 6th number. \[ \text{Median} = 225 \text{ pounds}\]
04

Identify the Mode

The mode is the number that appears most frequently in the list. Here, 190 and 305 appear twice each. \[ \text{Mode} = 190 \text{ and } 305 \text{ pounds}\]
05

Calculate the Midrange

The midrange is the average of the maximum and minimum values in the list. \[ \text{Midrange} = \frac{189 + 305}{2} \]\[ \text{Midrange} = \frac{494}{2} \]\[ \text{Midrange} = 247 \text{ pounds}\]
06

Analyzing Representation

To determine if these results are representative of all NFL players, consider the following factors: - **Sample Size**: The sample size of 11 players is relatively small compared to the entire league.- **Team Bias**: The weights are from players of a single team and may not represent the full diversity of player weights across all teams.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
The mean, or average, is a common way to describe the central tendency of a dataset. To find it, you need to sum all the values and then divide by the number of values. In this exercise, you have the weights of 11 Seattle Seahawks players. First, you sum the weights: 189 + 190 + 190 + 195 + 202 + 225 + 235 + 252 + 254 + 305 + 305 = 2742 pounds. Next, you divide by the number of players: 2742 / 11 which gives you about 249.27 pounds. This tells you that, on average, the players weigh about 249 pounds.
Median Finding
The median is the middle value in a sorted list of numbers. It provides a different measure of central tendency and is especially useful when your data contains outliers or skewed values. For 11 players, the median is the 6th weight in your ordered list. Arranging the weights: 189, 190, 190, 195, 202, 225, 235, 252, 254, 305, 305, we see the 6th number is 225 pounds. Thus, the median weight of these players is 225 pounds, which means half of the players weigh less than 225 pounds and half weigh more.
Mode Identification
The mode is the number that appears most frequently in a dataset. This can be useful for understanding the most common value in your data. For the weights provided, both 190 and 305 appear twice, while other weights appear only once. Hence, this list has two modes, 190 and 305 pounds. It's called a bimodal distribution since it has two modes. If only one value had appeared most frequently, the dataset would be referred to as unimodal.
Midrange Calculation
The midrange is a measure of central tendency that is computed by averaging the maximum and minimum values in a dataset. It provides a quick estimate of the midpoint. To compute it, you add the smallest weight (189 pounds) and the largest weight (305 pounds) together, then divide by two: \( \frac{189 + 305}{2} = 247 \). So, the midrange of these weights is 247 pounds. This measure can be influenced by extremely high or low values, so it鈥檚 important to consider whether it truly represents the center of your data.
Statistical Representation
When analyzing statistical data, it鈥檚 crucial to consider how representative the results are of the entire population. In this exercise, the data comes from a sample of only 11 players from one team.
  • Sample Size: A sample of 11 is relatively small compared to the total number of players in the NFL, which can affect the reliability of your statistics.
  • Team Bias: The weights are from just players on the Seattle Seahawks, which may not reflect the diversity of player weights across all teams in the NFL.
To better understand the representativeness, larger and more varied samples would give a more accurate reflection of all NFL players' weights.

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Most popular questions from this chapter

Use \(z\) scores to compare the given values. In the 87th Academy Awards, Eddie Redmayne won for best actor at the age of 33 and Julianne Moore won for best actress at the age of 54 . For all best actors, the mean age is 44.1 years and the standard deviation is \(8.9\) years. For all best actresses, the mean age is \(36.2\) years and the standard deviation is \(11.5\) years. (All ages are determined at the time of the awards ceremony.) Relative to their genders, who had the more extreme age when winning the Oscar: Eddie Redmayne or Julianne Moore? Explain.

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Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.) a. Find the variance \(\sigma^{2}\) of the population \(\\{9\) cigarettes, 10 cigarettes, 20 cigarettes \(\\}\). b. After listing the nine different possible samples of two values selected with replacement, find the sample variance \(s^{2}\) (which includes division by \(n-1\) ) for each of them; then find the mean of the nine sample variances \(s^{2}\). c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by \(n\) ); then find the mean of those nine population variances. d. Which approach results in values that are better estimates of \(\sigma^{2}\) : part (b) or part (c)? Why? When computing variances of samples, should you use division by \(n\) or \(n-1\) ? e. The preceding parts show that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\). Is \(s\) an unbiased estimator of \(\sigma\) ? Explain.

Use the following cell phone airport data speeds (Mbps) from Sprint. Find the percentile corresponding to the given data speed. $$\begin{array}{llllllllll} 0.2 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.4 & 0.4 & 0.4 \\ 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.6 & 0.6 & 0.7 & 0.8 & 1.0 \\ 1.1 & 1.1 & 1.2 & 1.2 & 1.6 & 1.6 & 2.1 & 2.1 & 2.3 & 2.4 \\ 2.5 & 2.7 & 2.7 & 2.7 & 3.2 & 3.4 & 3.6 & 3.8 & 4.0 & 4.0 \\ 5.0 & 5.6 & 8.2 & 9.6 & 10.6 & 13.0 & 14.1 & 15.1 & 15.2 & 30.4 \end{array}$$ \(9.6 \mathrm{Mbps}\)

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