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

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

(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.

(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.

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.

(c)

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

The frequency distributions for the net worth for celebrities are:

The highest frequency is obtained for 150.

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

(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.

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.

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.