Question

Critical Thinking. For Exercises 5–20, 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

Football Player Weights 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? 189 254 235 225 190 305 195 202 190 252 305

Short answer

(a) Mean: 231.1 pounds

(b) Median: 225.0 pounds

(c) Bimodal: 190 and 305 pounds

(d) Midrange: 247.0 pounds

The results are not likely to represent the population of all NFL players as the sample size is small and not drawn from a single team.

Step-by-step solution

Given information

The data for weights in pounds for 11 randomly chosen players is given below.

189, 254, 235, 225, 190, 305, 195, 202, 190, 252, 305

The players are selected from the roster of Seattle Seahawks created when they had won Super Bowl XLVIII.

Compute mean

(a).

The formula of mean is:

$\overline{x}=\frac{\sum x}{n}$, where xrepresents the observations in the sample and nis the total observations.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{189+254+...+305}{11} \\ =\frac{2542}{11} \\ \approx 231.0909 \end{gathered}$$

Thus, the mean value is approximately 231.1 pounds.

Compute median

(b).

The steps to compute median are:

• Arrange the data in a sorted manner.
• In case n is odd, the median is equal to the middlemost observation.
• In case n is even, the median is equal to the average of the two middle values.

The number of observations is 11.

Arrange the observations in ascending order.

189
190
190
195
202
225
235
252
254
305
305

The middlemost observation is 225.

The median is given as:

$M=225$

Thus, the median is 225.0 pounds.

Compute mode

(c).

The observations with the maximum repetitions are equal to mode.

The frequency distributions for the weights are:

Weights	Frequency
189	1
190	2
195	1
202	1
225	1
235	1
252	1
254	1
305	2

The highest frequency is obtained for two values. Thus, the data is bimodal, with the mode at 190 pounds and 305 pounds.

Compute midrange

(d)

The midrange is computed as:

$\text{Midrange}=\frac{\text{Minimum}\text{value}+\text{Maximum}\text{value}}{2}$

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{189+305}{2} \\ =\frac{494}{2} \\ =247 \end{gathered}$$

Thus, the midrange is 247.0 pounds.

Discuss if the results are representative of the population

The population includes all National Football League (NFL) players.

The players in the sample were selected from the roster of the Seattle Seahawks when they had won Super Bowl XLVIII.

The sample is gathered from an appropriate source through a simple random sampling procedure,although it is not of the appropriate size ( not large enough) and the sample is chosen from one team. Thus, the sample is not considered as representative of the population.

As a result, the results are not representative of all NFL players.