Problem 17 - Algebra 2

Question

An electronics store placed an ad in the newspaper showing five flat-screen TVs for sale. The ad says, "Our flat-screen TVs average \(\$ 695 . "\) The prices of the flat- screen TVs are \(\$ 1200, \$ 999, \$ 1499, \$ 695, \$ 695, \$ 1100, \$ 1300,\) and \(\$ 695 .\) Which measure is the store using in its ad? Why did they choose it?

Step-by-Step Solution

Answer

The store used the mode (\$695) in its ad because it appears most often and is the lowest price, making it more appealing to customers.

Step 1: Determine the possible measures of central tendency

The common measures of central tendency are mean, median, and mode. The mean is the average of the data set, the median is the middle value when the data points are ordered, and the mode is the most frequent data point.

Step 2: Calculate the mean price

To find the mean, sum all the TV prices and divide by the number of prices. The sum is \(1200 + 999 + 1499 + 695 + 695 + 1100 + 1300 + 695 = 8183\). There are 8 prices, so the mean is \(\frac{8183}{8} = 1022.875\).

Step 3: Calculate the median price

First, order the prices: \(695, 695, 695, 999, 1100, 1200, 1300, 1499\). There are 8 values, so the median is the average of the 4th and 5th values: \(\frac{999 + 1100}{2} = 1049.5\).

Step 4: Determine the mode of the prices

The mode is the value that appears most frequently. In the ordered list \(695, 695, 695, 999, 1100, 1200, 1300, 1499\), the number 695 appears three times, so 695 is the mode.

Step 5: Identify the measure used in the ad

The ad claims an average price of \\(695. Considering the calculations, the mode is \\)695. Therefore, the measure used in the ad is the mode.

Step 6: Reason for choosing the mode

The store likely chose the mode as it reflects the lowest and most appealing price to attract customers, giving a perception of more affordable pricing, although the average and median prices are significantly higher.

Key Concepts

Mean Calculation

The mean, often referred to as the average, is a fundamental concept in statistics used to determine the central tendency of a numerical dataset. To calculate the mean, you add up all the numbers in your dataset and then divide by the number of observations. This gives you a single value that represents the entire dataset.
For the given exercise, the TV prices are

• \(1200, \)999, \(1499, \)695, \(695, \)1100, \(1300, and \)695

The sum of these prices is $8183. Since there are 8 TVs, we divide the total sum by 8, resulting in a mean of \[\frac{8183}{8} = 1022.875\]This number represents the average price of the TVs. However, this mean is higher than the price advertised, indicating that the mean was not the measure used in the ad.

Median Calculation

The median is the middle value of a dataset when it's arranged in ascending order. It provides a measure of the center that is less affected by extremely large or small values, known as outliers.
To find the median in the list of TV prices, we first arrange them from smallest to largest:

• \(695, \)695, \(695, \)999, \(1100, \)1200, \(1300, \)1499

Since there are 8 prices, we want the average of the 4th and 5th values in this ordered list. Thus, the median is calculated as\[\frac{999 + 1100}{2} = 1049.5\]This median price reflects the middle ground of the dataset, yet it still does not align with the advertised price, reinforcing that the median was not used in the advertisement.

Mode Identification

The mode of a dataset is the value that appears most frequently. It is a useful measure of central tendency, particularly for categorical data, and provides insight into the most common item in a dataset.
For the TV prices, the list in ascending order is:

• $695, $695, $695, $999, $1100, $1200, $1300, $1499

Here, $695 appears three times, more than any other price. Hence, $695 is identified as the mode.
In the context of the advertisement, the store opted to promote $695 as the average price by using the mode, which was most frequent, making it appear as the typical price despite the mean and median being much higher.

Advertisement Analysis

Advertisement strategies often harness statistical measures that present data in the most favorable light. In this case, the store used the mode, advertising the most frequently occurring price, $695, as the 'average'.
This tactic can be appealing to consumers since it suggests affordability. However, this may not fully represent the pricing landscape since both mean ($1022.875) and median ($1049.5) are significantly higher.
This choice of using the mode helps in putting forth the lowest frequent price as typical, aligning product perceptions with price attractiveness. Hence, understanding the underlying statistics in such advertisements can provide consumers with a clearer picture of pricing scenarios.