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.

TV Prices Listed below are selling prices (dollars) of TVs that are 60 inches or larger and rated as a “best buy” by Consumer Reports magazine.

Are the resulting statistics representative of the population of all TVs that are 60 inches and larger?

If you decide to buy one of these TVs, what statistic is most relevant, other than the measures of central tendency?

1800 1500 1200 1500 1400 1600 1500 950 1600 1150 1500 1750

Short answer

(a) The mean is $1454.2.

(b) The median is $1500.0.

(c) The mode is $1500.

(d) The midrange is $1375.0.

The statistics are not considered to be representative of the population.

The lowest possible selling price must be known for better comparison.

Step-by-step solution

Given information

The selling prices for TVs larger than 60 inches listed as the best buy are recorded in dollars.

1800	1500	1200	1500	1400	1600
1500	950	1600	1150	1500	1750

Compute mean

(a)

The mean for a set of sample observations is computed as:

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

Here, xrepresents the observations and n is the sample size.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{1800+1500+...+1750}{12} \\ =\frac{17450}{12} \\ =454.167 \\ \approx 1454.2 \end{gathered}$$

Thus, the mean value is approximately $1454.2.

Compute median

(b)

The median for a set of observations is computed as:

For an odd number of observations, the middlemost observation is the median.

For an even number of observations, the average of the two middlemost observations is the median.

The sample size is 12.

Arrange the observations in ascending order.

950
1150
1200
1400
1500
1500
1500
1500
1600
1600
1750
1800

The middlemost observations are1500 and 1500.

The median is given as:

$$\begin{gathered} M=\frac{1500+1500}{2} \\ =1500 \end{gathered}$$

Thus, the median is $1500.0.

Compute mode

(c)

Mode is equal to the observation with the maximum frequency.

The frequency distributions for the prices are:

Prices in $	Frequency
950	1
1150	1
1200	1
1400	1
1500	4
1600	2
1750	1
1800	1

The highest frequency is obtained for $1500.

Thus, the mode of the data is $1500.

Compute midrange

(d)

The midrange for a set of observations is computed as:

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

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{950+1800}{2} \\ =\frac{2750}{2} \\ =1375 \end{gathered}$$

Thus, the midrange is $1375.0.

Discuss if the statistics are representative of the population

The sample consists of data from only the best buy TVs of the required dimension. The sample is, therefore, not an appropriate representative of the population of all TVs.

Thus, the resultant statistics are not representative of the population.

Discuss the most relevant statistics

From a buyer’s perspective, it is important to identify the lowest selling price of the TV set of the specific dimension for better comparison.