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Chapter 5: Problem 14

The prices of four steam irons are \( 50, \) 30, \( 25,\) and $$ 45\( a. Find the mean of these prices. b. An iron that costs $$ 140\) is added to the list. What is the mean of all five irons? c. How does the expensive iron affect the mean?

Short Answer

Expert verified
a. 37.5 b. 58 c. The mean increased.

Step by step solution

01

- Calculate the Mean of Four Irons

The mean price of the four steam irons can be found by summing their prices and dividing by the number of irons. The prices are 50, 30, 25, and 45.Sum of prices = 50 + 30 + 25 + 45 Finding the mean, we divide this total by 4 since there are four irons.Mean = \( \frac{50 + 30 + 25 + 45}{4} \)
02

- Calculate the Sum

First, calculate the sum of the initial prices:\( 50 + 30 + 25 + 45 = 150 \).So, the mean = \( \frac{150}{4} = 37.5 \). The mean of the four irons is 37.5.
03

- Add the New Iron Price

Now include the additional iron priced at 140. The new list of prices is 50, 30, 25, 45, and 140.We need to find the sum of these prices and then divide by the new number of irons, which is 5.
04

- Calculate the New Sum

Sum of prices including the new iron = 50 + 30 + 25 + 45 + 140.This equals = 290. So, the new mean = \( \frac{290}{5} = 58 \).
05

- Compare the Means

Compare the original mean (37.5) with the new mean (58). The addition of the expensive iron increased the mean price of the irons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

average
The concept of an average is crucial in various areas, especially in mathematics and statistics. An average, also known as the mean, is a measure that represents the central tendency of a set of numbers. It is obtained by summing up all the numbers in a given set and then dividing the total by the count of numbers. For instance, if the prices of four steam irons are 50, 30, 25, and 45, you first add these prices: 50 + 30 + 25 + 45. This gives you a sum of 150. Then, you divide this sum by 4 (since there are four prices), giving you 37.5. This value, 37.5, is the average price of the four steam irons.
mean price
When discussing the cost of products, the term 'mean price' often comes up. The mean price is simply the average cost of those items. For the initial set of four steam irons, the mean price is calculated by summing up all their prices (50 + 30 + 25 + 45 = 150) and then dividing by the number of items (4). So, the mean price = \( \frac{150}{4} = 37.5 \). However, when a new, more expensive iron priced at 140 is added to the list, the total sum of prices changes to 290. By dividing this new sum by the total number of irons (5), the new mean price becomes \( \frac{290}{5} = 58 \). The mean price, in this case, increased from 37.5 to 58.
impact of outliers
Outliers are values that are significantly higher or lower than most of the values in a data set. They can have a considerable impact on the mean. In our example, the prices of the steam irons were initially 50, 30, 25, and 45, with a calculated mean price of 37.5. Adding an expensive iron priced at 140 drastically changes the mean price to 58. This indicates the influence of the outlier (140) on the new average. Outliers can sometimes provide valuable insights, but they can also skew the mean, making it less representative of the general data set.
basic arithmetic operations
Calculating the mean or average requires a good grasp of basic arithmetic operations like addition and division. To determine the mean, you first add all the given numbers and then divide the resulting sum by the number of items. For instance, with the given prices of steam irons: 50, 30, 25, 45, and later adding 140, you start by finding their sum (50 + 30 + 25 + 45 + 140 = 290). You then divide this total sum by the count of the items, which is 5. Hence, the mean price = \( \frac{290}{5} = 58 \). Mastering these operations helps you easily calculate averages in various real-life contexts.

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