Question

find the (a) mean, (b) median, (c) mode, and (d) midrange for the data and then (e) answer the given questions.
listed below are selling prices (dollars) of tvs that are 60 inches or larger and rated as a \best buy\ by a popular 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 1650 1100 1650 1950 1750 1850 1550 1750 1300 1650 1600

b. find the median.
(type an integer or a decimal rounded to one decimal place as needed.)
the median is $ 1650.

c. find the mode.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the mode(s) is(are) $ 1650.
(type an integer or a decimal. do not round. use a comma to separate answers as needed.)
b. there is no mode.

d. find the midrange.
(type an integer or a decimal rounded to one decimal place as needed.)
the midrange is $ 1525.

e. are the resulting statistics representative of the population of all tvs that are 60 inches and larger? choose the best answer below.
a. the sample consists of the \best buy\ tvs, so it is not a random sample and is not likely to be representative of the population.
b. since the sample is random and the sample size is greater than 10, the sample can be considered to be representative of the population.
c. since the sample is random and the sample size is greater than 10, the sample should not be considered to be representative of the population.
d. the sample consists of the \best buy\ tvs, so it is a random sample and is likely to be representative of the population.

Explanation:

Part (a): Find the Mean

Step 1: Sum all the data values

First, we list out the data values: \(1800, 1650, 1100, 1650, 1950, 1750, 1850, 1550, 1750, 1300, 1650, 1600\).
Let's sum them up:
\[

$$\begin{align*} &1800 + 1650 + 1100 + 1650 + 1950 + 1750 + 1850 + 1550 + 1750 + 1300 + 1650 + 1600\\ =& (1800 + 1650) + 1100 + (1650 + 1950) + (1750 + 1850) + (1550 + 1750) + (1300 + 1650) + 1600\\ =& 3450 + 1100 + 3600 + 3600 + 3300 + 2950 + 1600\\ =& 4550 + 3600 + 3600 + 3300 + 2950 + 1600\\ =& 8150 + 3600 + 3300 + 2950 + 1600\\ =& 11750 + 3300 + 2950 + 1600\\ =& 15050 + 2950 + 1600\\ =& 18000 + 1600\\ =& 19600 \end{align*}$$

\]

Step 2: Divide the sum by the number of data points

There are \(n = 12\) data points. The mean \(\bar{x}\) is given by the formula \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\).
So, \(\bar{x}=\frac{19600}{12}\approx1633.3\) (rounded to one decimal place)

Step 1: Order the data set

First, we order the data from smallest to largest: \(1100, 1300, 1550, 1600, 1650, 1650, 1650, 1750, 1750, 1800, 1850, 1950\)

Step 2: Find the middle value(s)

Since \(n = 12\) (even), the median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+ 1)\)-th values.
\(\frac{n}{2}=\frac{12}{2}=6\) and \(\frac{n}{2}+1 = 7\)
The 6 - th value is \(1650\) and the 7 - th value is \(1650\)
The median \(=\frac{1650 + 1650}{2}=1650\)

Step 1: Identify the frequency of each value

• \(1100\): 1 time
• \(1300\): 1 time
• \(1550\): 1 time
• \(1600\): 1 time
• \(1650\): 3 times
• \(1750\): 2 times
• \(1800\): 1 time
• \(1850\): 1 time
• \(1950\): 1 time

Step 2: Determine the mode

The mode is the value with the highest frequency. Since \(1650\) appears 3 times, which is more than any other value.

Answer:

The mean is \(\$1633.3\)

Part (b): Find the Median