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.

California Smokers In the California Health Interview Survey, randomly selected adults are interviewed. One of the questions asks how many cigarettes are smoked per day, and results are listed below for 50 randomly selected respondents. How well do the results reflect the smoking behavior of California adults?

9 10 10 20 40 50 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Short answer

(a) The mean is 2.8 cigarettes.

(b) The median is 0.0 cigarettes.

(c) The mode is 0 cigarettes.

(d) The midrange is 25.0 cigarettes

The data is not appropriate as the observations are reported, not observed.

Step-by-step solution

Given information

The responses for the number of cigarettes smoked in a day are recorded from randomly selected respondents.

9, 10, 10, 20, 40, 50, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Compute mean

(a)

The mean of observations is calculated as follows:

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

Here, xis the observations, and nis the count of the observations.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{9+10+10+...+0}{50} \\ =\frac{139}{50} \\ \approx 2.78 \end{gathered}$$

Thus, the mean value is approximately 2.8 cigarettes.

Compute median

(b)

Median is a specific value that is obtained as follows:

• n is even; mean of two middle observations.
• n is odd; the middle observation.

The number of observations is50.

Arrange the observations in ascending order.

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
9
10
10
20
40
50

The middlemost observations are 0 and 0.

The median is given as:

$$\begin{gathered} M=\frac{0+0}{2} \\ =0 \end{gathered}$$

Thus, the median is 0.0 cigarettes.

Compute mode

(c)

The value which repeats the most is mode.

The frequency distribution is:

Observations	Frequency
0	44
9	1
10	2
20	1
40	1
50	1

As 0 repeats the most, the mode is 0.

Thus, the mode is 0 cigarettes.

Compute midrange

(d)

The midrange is determined by the following formula as the average of extreme values:

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

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{0+50}{2} \\ =\frac{50}{2} \\ =25 \end{gathered}$$

Thus, the midrange is 25.0 cigarettes.

Express the smoking behavior from the data

The respondents were randomly selected and self-reported the number of cigarettes smoked. It is highly likely that they would report incorrect numbers as they were not observed for the same. Also, it is unlikely that an individual smokes 40 or 50 cigarettes in one day.

Therefore, the data can be inaccurate.

Thus, it can be concluded that the data may not be an accurate representation of the population of all adults in California. Consequently, the results are not a correct representation of the population.