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Chapter 6: Problem 28

An interesting use of the inclusion/exclusion rule is to check survey numbers for consistency. For example, suppose a public opinion polltaker reports that out of a national sample of 1,200 adults, 675 are married, 682 are from 20 to 30 years old, 684 are female, 195 are married and are from 20 to 30 years old, 467 are married females, 318 are females from 20 to 30 years old, and 165 are married females from 20 to 30 years old. Are the polltaker's figures consistent? Could they have occurred as a result of an actual sample survey?

Short Answer

Expert verified
Using the inclusion-exclusion principle, we can determine if the given data is consistent. We calculate |A 鈭� B 鈭� C| = 675 + 682 + 684 - 195 - 467 - 318 + 165, which equals the total number of adults surveyed (1200). Since the values match, the polltaker's figures are consistent and could have occurred as a result of an actual sample survey.

Step by step solution

01

Apply the inclusion-exclusion principle formula: |A 鈭� B 鈭� C| = |A| + |B| + |C| - |A 鈭� B| - |A 鈭� C| - |B 鈭� C| + |A 鈭� B 鈭� C| Using the given data, we can substitute the corresponding values: |A 鈭� B 鈭� C| = 675 + 682 + 684 - 195 - 467 - 318 + 165 #Step 2: Check for consistency#

Calculate the total number of adults surveyed using the inclusion-exclusion principle: |A 鈭� B 鈭� C| = 675 + 682 + 684 - 195 - 467 - 318 + 165 = 1200 Since the calculated value of |A 鈭� B 鈭� C| is equal to the total number of adults surveyed (1200), the polltaker's figures are consistent and could have occurred as a result of an actual sample survey.

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Key Concepts

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

Set Theory
The Inclusion-Exclusion Principle is a fundamental aspect of set theory, which deals with the study of collections of objects, referred to as sets. In our context, set theory helps us understand how overlapping groups combine in a survey. To grasp the core idea, think of the principle as a way to accurately calculate the size of the union of several sets. When individuals belong to more than one group, simply adding the sizes of each group overestimates the total number. This principle corrects the overestimation by subtracting the overlaps and then adding back the overlaps that have been subtracted too many times.

This becomes crucial when working with multiple sets like married adults, adults aged 20 to 30, and female adults. Each set overlaps in the survey, leading to a union of three groups, and we must apply the inclusion-exclusion principle to ensure we do not miscalculate the total.
Survey Analysis
To achieve mathematical consistency in survey results using the Inclusion-Exclusion Principle, it's essential to ensure that a survey's reported figures accurately reflect possible real-world scenarios. Survey analysis involves examining data to find patterns and insights. It's about checking if the data logically fits together and can be expected from a randomly chosen sample.

When analyzing survey data, consider:
  • The total number of participants, which impacts all calculations.
  • Double-counting individuals who belong to multiple categories.
  • Ensuring that any derived total matches the expected sample size.
By understanding these points, you can verify whether a polling report is consistent with actual possibilities. Thus, by using accurate calculations, the survey analysis remains reliable and insightful.
Mathematical Consistency
Mathematical consistency ensures that all calculations in a survey add up logically and adhere to mathematical rules. This concept is essential to verifying the accuracy of data and its representation.

In mathematical consistency, each set in our survey represents a different group of individuals: married, aged 20 to 30, and female. Overlapping sets must be treated carefully to avoid mishandling numbers. By applying the Inclusion-Exclusion Principle and reaching a total that matches the surveyed population, you maintain consistency.

To ensure that each number is accurate, you need to:
  • Verify overlaps are correctly subtracted and added back.
  • Match calculated expectations with the actual sample size.
  • Adjust any discrepancies, ensuring the mathematical accuracy of your results.
This process affirms that the survey figures aren't just random or flawed, but are results that could feasibly emerge from an appropriately conducted survey.

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Most popular questions from this chapter

Two faces of a six-sided die are painted red, two are painted blue, and two are painted yellow. The die is rolled three times, and the colors that appear face up on the first, second, and third rolls are recorded. a. Let \(B B R\) denote the outcome where the color appearing face up on the first and second rolls is blue and the color appearing face up on the third roll is red. Because there are as many faces of one color as of any other, the outcomes of this experiment are equally likely. List all 27 possible outcomes. b. Consider the event that all three rolls produce different colors. One outcome in this event is \(R B Y\) and another \(R Y B\). List all outcomes in the event. What is the probability of the event? c. Consider the event that two of the colors that appear face up are the same. One outcome in this event is \(R R B\) and another is \(R B R\). List all outcomes in the event. What is the probability of the event?

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Express each of the sums in \(24-35\) in closed form (without using a summation symbol and without using an ellipsis \(\cdots\) ). $$ \sum_{i=0}^{m}\left(\begin{array}{c} m \\ i \end{array}\right) p^{m-i} q^{2 i} $$

If \(n\) is a positive integer, how many 4-tuples of integers from 1 through \(n\) can be formed in which the elements of the 4-tuple are written in increasing order but are not necessarily distinct? In other words, how many 4-tuples of integers \((i, j, k, m)\) are there with \(1 \leq i \leq j \leq k \leq m \leq n ?\)
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