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

Listed below are the highest amounts of net worth (in millions of dollars) of celebrities. The celebrities are Tom Cruise, Will Smith, Robert De Niro, Drew Carey, George Clooney, John Travolta, Samuel L. Jackson, Larry King, Demi Moore, and Bruce Willis.

What do the results tell us about the population of all celebrities? Based on the nature of the amounts, what can be inferred about their precision?

250 200 185 165 160 160 150 150 150 150

Short answer

(a). The mean is $172.0 million.

(b). The median is $160.0 million.

(c). The mode is $150 million.

(d). The midrange is $200.0 million.

No useful inference can be derived as the sampling method is unknown. Also, the amounts are not the precise amounts of net worth but a rounded off estimate.

Step-by-step solution

Given information

The highest net worth for ten celebrities are known in millions of dollars and it

is listed below,

250, 200, 185, 165, 160, 160, 150, 150, 150, 150

Compute mean

(a)

The sample mean is computed using the following formula.

$\overline{x}=\frac{\sum x}{n}$

Here,

• x represents the sample observations
• n is the count of the sample observations.

The number of observations is 10.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{250+200+...+150}{10} \\ =\frac{1720}{10} \\ =172 \end{gathered}$$

Thus, the mean value is approximately $172.0 million.

Compute median

(b)

The median is obtained from the data arranged in ascending order as follows:

• When the number of observations is odd, the median is the middlemost observation.
• When the number of observations is even, the median is the average of the two middlemost observations.

Arrange the observations in ascending order.

150
150
150
150
160
160
165
185
200
250

The middlemost observations are 160 and 160.

The median is given as:

$$\begin{gathered} M=\frac{160+160}{2} \\ =160 \end{gathered}$$

Thus, the median is $160.0 million.

Compute mode

(c)

Mode is(are) the observation(s) with the maximum frequency.

The frequency distributions for the net worth for celebrities are:

Net worth	Frequency
150	4
160	2
165	1
185	1
200	1
250	1

The highest frequency is obtained for 150.

Thus, the mode of the data is $150 million.

Compute midrange

(d)

The midrange is determined as follows:

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

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{150+250}{2} \\ =\frac{400}{2} \\ =200 \end{gathered}$$

Thus, the midrange is $200.0 million.

Discuss the results with reference to the population of the celebrities

The data was recorded for the highest net worths for 10 sampled celebrities. As the sampling method for selecting the celebrities is unknown, the results cannot be accurately mapped for the population of all celebrities.

Moreover, the celebrities mentioned in the sample are well-known and most popular in the industry. Thus, it can be inferred that the other celebrities would have a net worth lower than the sampled amounts.

Discuss the precision of the sampled amounts

As all amounts are rounded off to a whole number, it can be inferred that these are not actual values but a rounded off estimate of the actual highest net worth of the celebrities.