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Chapter 14: Problem 34

According to the U.S. Census Bureau, in \(2012,\) about \(68 \%\) (.68) of children in the United States were living with both parents, \(24.4 \%\) (.244) were living with mother only, \(4 \%\) (.04) were living with father only, and \(3.6 \%\) (.036) were not living with either parent. What is the expected value for the number of parents a randomly selected child was living with? Does the concept of expected value have a meaningful interpretation for this example? Explain.

Short Answer

Expert verified
The expected value is 1.644. It reflects the average number of parents a child lived with and serves as a measure of central tendency.

Step by step solution

01

Understand the Problem

We need to find the expected value for the number of parents a child was living with, based on given probabilities. This involves calculating a weighted average of the number of parents a child is living with, using probabilities as weights.
02

Set Up a Probability Table

Convert the percentage of children into probabilities: \( P(2 \text{ parents}) = 0.68 \), \( P(1 \text{ parent, mother only}) = 0.244 \), \( P(1 \text{ parent, father only}) = 0.04 \), and \( P(0 \text{ parents}) = 0.036 \).
03

Define Random Variable X

Define \( X \) as the number of parents a child is living with. The possible values of \( X \) are: \( X = 0 \), \( X = 1 \), or \( X = 2 \), with corresponding probabilities assigned to each outcome.
04

Calculate Expected Value

Use the formula for expected value: \[ E(X) = (2 \cdot P(2 \text{ parents})) + (1 \cdot (P(1 \text{ parent, mother only}) + P(1 \text{ parent, father only}))) + (0 \cdot P(0 \text{ parents})). \]Substituting in the probabilities, we have: \[ E(X) = (2 \times 0.68) + (1 \times (0.244 + 0.04)) + (0 \times 0.036) \] \[ = 1.36 + 0.284 + 0 \] \[ = 1.644. \]
05

Interpret the Expected Value

The expected value, 1.644, represents the average number of parents a randomly selected child was living with in 2012. While no child can literally live with 1.644 parents, this value provides a statistical measure of central tendency, or what you might typically expect when considering a whole population.

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Key Concepts

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

Probability Distribution
Probability distribution is a concept that describes how probabilities are assigned to different potential outcomes of a random process. In our specific example, the probabilities are related to different scenarios of a child living with a certain number of parents.
  • There are four different living situations a child could be in: living with both parents, with mother only, with father only, or not with either parent.
  • Each of these scenarios is assigned a specific probability that all together add up to 1, or 100%.
  • The probability distribution helps us understand the likelihood of each possible outcome.
In essence, the probabilities reflect the distribution of children across these various living conditions.
Weighted Average
A weighted average is a type of calculation that takes into account not just the values in a dataset but also how often each value occurs. In our case, the number of parents is weighted by the probability of each situation.
  • The child's possible number of parents can be 0, 1, or 2.
  • Each situation's probability acts as a weight, influencing the average.
  • It's calculated by multiplying each value by its probability and then adding those products together.
This form of averaging accounts for the varying significance of each outcome, providing a more accurate and practical representation of the dataset.
Random Variable
In statistics and probability, a random variable is a numerical value determined by the outcome of a random event. Here, the random variable, denoted as \(X\), represents the number of parents a child lives with.
  • Random variables can take on different values: for a child, this could be 0, 1, or 2 parents.
  • Each value of the random variable has an associated probability.
  • The idea of the random variable allows us to assign numerical values to outcomes of random processes, aiding in quantitative analysis.
This means, instead of dealing with abstract concepts, we use \(X\) for clearer and more structured exploration of the problem.
Statistical Interpretation
Statistical interpretation involves understanding what numerical outcomes mean in context. The expected value calculated in our problem is 1.644, which suggests the number of parents a child lives with on average.
  • The expected value provides insight into the central tendency of data, indicating a typical outcome.
  • While it isn't a valid number of parents a child could have, it symbolizes an average across a population of varying conditions.
  • This measure helps in comparing different populations or assessing societal trends.
The analysis bridges numbers with real-world interpretation, offering a tool for understanding larger patterns and deviations within the surveyed population.

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Most popular questions from this chapter

Suppose the probability that you get an interesting piece of mail on any given weekday is \(1 / 20 .\) Is the probability that you get at least one interesting piece of mail during the week (Monday to Friday) equal to \(5 / 20 ?\) Why or why not?

Use the probability rules in this chapter to solve each of the following: a. According to the U.S. Census Bureau, in 2012 , the probability that a randomly selected child in the United States was living with his or her mother as the sole parent was .244 and with his or her father as the sole parent was. \(040 .\) What was the probability that a child was living with just one parent? b. In 2010 in the United States, the probability that a birth would result in twins was \(.0331,\) and the probability that a birth would result in triplets or more was .0014. What was the probability that a birth in 2010 resulted in a single child?

People are surprised to find that it is not all that uncommon for two people in a group of 20 to 30 people to have the same birthday. We will learn how to find that probability in a later chapter. For now, consider the probability of finding two people who have birthdays in the same month. Make the simplifying assumption that the probability that a randomly selected person will have a birthday in any given month is \(1 / 12 .\) Suppose there are three people in a room and you consecutively ask them their birthdays. Your goal, following parts (a-d), is to determine the probability that at least two of them were born in the same calendar month. a. What is the probability that the second person you ask will not have the same birth month as the first person? (Hint: Use Rule 1.) b. Assuming the first and second persons have different birth months, what is the probability that the third person will have yet a different birth month? (Hint:Suppose January and February have been taken. What proportion of all people will have birth months from March to December?) c. Explain what it would mean about overlap among the three birth months if the outcomes in part (a) and part (b) both happened. What is the probability that the outcomes in part (a) and part (b) will both happen? d. Explain what it would mean about overlap among the three birth months if the outcomes in part (a) and part (b) did not both happen. What is the probability of that occurring?

On any given day, the probability that a randomly selected adult male in the United States drinks coffee is \(.51(51 \%),\) and the probability that he drinks alcohol is .31 On any given day, the probability that a randomly selected adult male in the United States drinks coffee is \(.51(51 \%),\) and the probability that he drinks alcohol is .31 (31\%). (Source: http://www.ars.usda.gov/SP2UserFiles/Place/12355000/pdf/DBrief/6 beverage choices adults 0708 . \(\mathrm{pdf}\) ) What assumption would we have to make in order to use Rule 3 to conclude that the probability that a person drinks both is(.51) \(\times(.31)=.158 ?\) Do you think that assumption holds in this case? Explain.

For each of the following situations, state whether the relative-frequency interpretation or the personal probability interpretation is appropriate. If it is the relative-frequency interpretation, specify which of the two methods for finding such probabilities would apply. a. If a spoon is tossed 10,000 times and lands with the rounded head face up on 3000 of those times, we would say that the probability of the rounded head landing face up for that spoon is about. 30. b. In a debate with you, a friend says that she thinks there is a \(50 / 50\) chance that God exists.
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