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Chapter 3: Problem 24

Basic Computation: Weighted Average Find the weighted average of a data set where 10 has a weight of \(5 ; 20\) has a weight of \(3 ; 30\) has a weight of 2

Short Answer

Expert verified
The weighted average is 17.

Step by step solution

01

Understanding the Concept

A weighted average is calculated by multiplying each number in the data set by its corresponding weight and then summing the results. The total sum is then divided by the sum of the weights.
02

Multiply Each Number by Its Weight

Calculate the multiplied values as follows: - For 10 with a weight of 5, compute: \( 10 imes 5 = 50 \) - For 20 with a weight of 3: \( 20 imes 3 = 60 \) - For 30 with a weight of 2: \( 30 imes 2 = 60 \)
03

Sum the Weighted Values

Add the results from Step 1 to get the total weighted value: \( 50 + 60 + 60 = 170 \)
04

Sum the Weights

Add all the weights together: \( 5 + 3 + 2 = 10 \)
05

Calculate the Weighted Average

To find the weighted average, divide the total weighted value by the sum of the weights: \( \frac{170}{10} = 17 \)

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Key Concepts

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

Data Set
The concept of a "Data Set" is foundational in statistics. In essence, a data set is a collection of related numbers or values that are often analyzed to draw conclusions or make informed decisions. Here, the data set comprises numbers like 10, 20, and 30, each linked to specific weights. Understanding how these numbers relate to one another is crucial before any calculations are carried out. By identifying each item in the data set, you can effectively determine how they collectively influence the final results. It's similar to looking at a group of building blocks 鈥� each block (or number) is distinct and how they stack up contributes to the overall structure (the result).
Weighted Values
Weighted values are integral to the concept of weighted averages. They represent the product of each value in the data set and its corresponding weight. For instance, if we take the number 10 and its weight of 5, the weighted value would be calculated as:
  • For 10: \(10 \times 5 = 50\)
  • For 20: \(20 \times 3 = 60\)
  • For 30: \(30 \times 2 = 60\)
These computations ensure that each number is considered according to its importance or frequency. By examining the individual contributions of each weighted value, it becomes clearer how each component impacts the average. It's like assigning different levels of importance to tasks in a list and then measuring their cumulative impact.
Calculation Method
The "Calculation Method" for finding a weighted average begins with an understanding of how to combine both values and weights. Each number in the data set is first multiplied by its corresponding weight to produce weighted values. The process can be broken down into straightforward steps:
  • Step 1: Multiply each number by its weight, resulting in weighted values.
  • Step 2: Summate these weighted values to attain a total weighted sum.
  • Step 3: Add up all the individual weights to find the sum of weights.
  • Step 4: Divide the total weighted sum by the sum of weights.
The final result is the weighted average, which leverages both data and weights effectively to deliver a balanced answer. This systematic approach ensures that each part of the calculation contributes to an accurate and meaningful average value.
Sum of Weights
"Sum of Weights" represents all individual weights added together. This sum is crucial because it acts as a divisor in the calculation of the weighted average. In the given exercise, we find this by simply adding the weights:
  • For weights 5, 3, and 2, the sum is: \(5 + 3 + 2 = 10\)
The sum of weights helps normalize the weighted sum by balancing out all the influences which are expressed through their respective weights. Without this step, the final average would not hold true significance. This part of the method ensures proportional representation of each weighted contribution, thus delivering a clear and unbiased weighted average value.

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Most popular questions from this chapter

Wolf Packs How large is a wolf pack? The following information is from a random sample of winter wolf packs in regions of Alaska, Minnesota, Michigan, Wisconsin, Canada, and Finland (Source: The Wolf, by L. D. Mech, University of Minnesota Press). Winter pack size: $$\begin{array}{ccccccccc}13 & 10 & 7 & 5 & 7 & 7 & 2 & 4 & 3 \\\2 & 3 & 15 & 4 & 4 & 2 & 8 & 7 & 8\end{array}$$ Compute the mean, median, and mode for the size of winter wolf packs.

Consider a data set with at least three data values. Suppose the highest value is increased by 10 and the lowest is decreased by \(5 .\) (a) Does the mean change? Explain. (b) Does the median change? Explain. (c) Is it possible for the mode to change? Explain.

Given the sample data \(\begin{array}{cccccccc}x: & 23 & 17 & 15 & 30 & 25\end{array}\) (a) Find the range. (b) Verify that \(\Sigma x=110\) and \(\Sigma x^{2}=2568\) (c) Use the results of part (b) and appropriate computation formulas to compute the sample variance \(s^{2}\) and sample standard deviation \(s\) (d) Use the defining formulas to compute the sample variance \(s^{2}\) and sample standard deviation \(s\) (e) Suppose the given data comprise the entire population of all \(x\) values. Compute the population variance \(\sigma^{2}\) and population standard deviation \(\sigma .\)

When computing the standard deviation, does it matter whether the data are sample data or data comprising the entire population? Explain.

Angela took a general aptitude test and scored in the \(82 n d\) percentile for aptitude in accounting. What percentage of the scores were at or below her score? What percentage were above?
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