Problem 18 - 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 \(\$ 695 .\) As a consumer, which measure would you want to see advertised? Explain.

Step-by-Step Solution

Answer

See the mode; it shows the most common price.

Step 1: Understand the Given Information

The problem provides the prices of eight flat-screen TVs: - $1200, $999, $1499, $695, $695, $1100, $1300, and $695. The advertisement claims that the average price of the TVs is $695.

Step 2: Calculate the Mean

To find the mean (average) price of the TVs, sum all the prices and divide by the number of TVs.\[ \text{Mean} = \frac{1200 + 999 + 1499 + 695 + 695 + 1100 + 1300 + 695}{8} \]\[ \text{Mean} = \frac{8183}{8} \approx 1022.88 \]The mean price is approximately \(1022.88, not the \)695 claimed in the ad.

Step 3: Calculate the Median

To find the median, arrange the prices in ascending order and find the middle value.Ordered prices: \(695, \)695, \(695, \)999, \(1100, \)1200, \(1300, \)1499.Since there are 8 prices, the median is the average of the 4th and 5th prices:\[ \text{Median} = \frac{999 + 1100}{2} = 1049.5 \]The median price is $1049.5.

Step 4: Identify the Mode

The mode is the price that appears most frequently. In the list: $695 appears 3 times, while other values appear only once or twice. Thus, the mode is $695.

Step 5: Determine the Most Relevant Statistic

As a consumer, seeing the mode is most relevant because it reflects the most frequently occurring price. This gives a better sense of what price is most often encountered and aligns with consumer perception.

Key Concepts

Mean

The mean is a fundamental concept in statistics. It represents the average value of a data set. To calculate the mean of a group of numbers, you sum them all up and then divide by the total quantity in the group.
In our flat-screen TV example, the sum of the prices is $8183, and there are 8 TVs. To find the mean, you divide $8183 by 8, which results in approximately $1022.88.
This mean price is significantly higher than the $695 advertised by the store. Calculating the mean helps consumers understand the typical price they are likely to encounter in the dataset.

• Sum up all values
• Divide by total number of values

Understanding mean provides insight into the general tendency of the prices.

Median

The median represents the middle value in a sorted list of numbers. It gives a sense of the central tendency in the data.
To find the median, first arrange the numbers in ascending order. For our TV prices, we order them as \(695, \)695, \(695, \)999, \(1100, \)1200, \(1300, \)1499.
Since there are 8 values (an even number), the median is the average of the 4th and 5th numbers: \(999 and \)1100. Calculate the median using the formula \[ \text{Median} = \frac{999 + 1100}{2} = 1049.5 \]
The median of $1049.50 is a more accurate reflection of the middle pricing, not subject to extreme values on either end.

• Arrange data in order
• If even number of data points, average the two middle numbers

Median offers a balanced view unaffected by outliers.

Mode

The mode is the value that appears most frequently in a dataset. It emphasizes what occurs most often.
In the case of our flat-screen TVs, the price $695 appears three times, while all other prices appear only once or twice. Therefore, $695 is the mode.
This might be why the store chose to advertise this as it reflects a common price among the products being sold.
The mode is particularly useful to show the most frequent price range in consumer studies.
Knowing the mode helps consumers quickly gauge the price point they are most likely to come across.

• Easy to identify in a dataset
• Reflects commonality in values

When assessing average costs, mode can offer practical market understanding.

Consumer Perception

Consumer perception revolves around how potential buyers view and interpret prices, products, and advertising.
When a store advertises the price of $695, they present the mode as the representative statistic because it reflects the price most frequently listed. This can influence consumer perception by suggesting this is the typical cost of the TVs.
In the context of shopping, buyers might favor lower prices or look for certain patterns.

• Signals common pricing
• Affects shopping behavior

However, relying solely on mode doesn't always paint a complete picture. Consumers should consider all statistical measures to make informed decisions about products.