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Chapter 6: Problem 45

The report "TV Drama/Comedy Viewers and Health Information" (www.cdc.gov/Healthmarketing) describes the results of a large survey involving approximately 3500 people that was conducted for the Center for Disease Control. The sample was selected in a way that the Center for Disease Control believed would result in a sample that was representative of adult Americans. One question on the survey asked respondents if they had learned something new about a health issue or disease from a TV show in the previous 6 months. Data from the survey was used to estimate the following probabilities, where \(L=\) event that a randomly selected adult American reports learning something new about a health issue or disease from a TV show in the previous 6 months and \(F=\) event that a randomly selected adult American is female $$ P(L)=.58 \quad P(L \cap F)=.31 $$ Assume that \(P(F)=.5\). Are the events \(L\) and \(F\) independent events? Use probabilities to justify your answer.

Short Answer

Expert verified
No, the events \(L\) and \(F\) are not independent, they are dependent events, as the equation \(P(L \cap F) = P(L)*P(F)\) does not hold true. This is because 0.31 \neq 0.29.

Step by step solution

01

Understand the Problem

The problem has given the probabilities \(P(L) = 0.58\) (the probability that an adult learned from a TV show), \(P(F) = 0.5\) (the probability that an adult is female) and \(P(L \cap F) = 0.31\) (the probability that an adult is female and learned from a TV show). The task is to determine if events \(L\) and \(F\) are independent.
02

Apply the definition of independence

Two events \(A\) and \(B\) are independent if the probability of both events occurring is the product of their individual probabilities, i.e., \(P(A \cap B) = P(A)*P(B)\). We need to check this for our events \(L\) and \(F\).
03

Calculate if \(P(L \cap F) = P(L)*P(F)\)

Substitute the provided probabilities into the formula from Step 2. We calculate \(P(L)*P(F) = 0.58*0.5 = 0.29\). Now compare this value with the given \(P(L \cap F) = 0.31\).
04

Determine if events are independent or dependent

Since \(P(L)*P(F) \neq P(L \cap F)\) (0.29 \neq 0.31), the events \(L\) and \(F\) are not independent. Hence, the events are dependent.

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Key Concepts

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

Probability
Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. When you hear percentages mentioned in daily life, such as a 30% chance of rain, you are encountering a concept rooted in probability. In statistics, probability ranges from 0 to 1, where 0 indicates the impossibility of an event, and 1 signifies certainty. For example, in the survey exercise provided, we have probabilities for different events.
  • Given: The probability that an individual learned from a TV show, represented as \( P(L) = 0.58 \).
  • The probability that an individual is female, represented as \( P(F) = 0.5 \).

These probabilities help us understand and predict behaviors in a population. The exact probabilities allow statisticians to infer insights and make predictions about larger populations based on these samples. By comparing these values, as seen in the original exercise, statisticians can gain insights into how certain events might relate to one another.
Independence of Events
Understanding the independence of events is vital in determining how two events are related. Two events, labeled as \( A \) and \( B \), are independent if the occurrence of one event does not affect the occurrence probability of the other.
Let's consider the events in a real-world scenario. For the events \( L \) and \( F \) in the survey:
  • Event \( L \) is learning something new from a TV show.
  • Event \( F \) is being female.
According to the formula of independence, \( P(A \cap B) = P(A) \cdot P(B) \), if this holds true, the events are independent. Applying this to our exercise:
  • Calculate \( P(L) \cdot P(F) = 0.58 \times 0.5 = 0.29 \).
  • Compare with \( P(L \cap F) = 0.31 \).

Since \( P(L) \cdot P(F) eq P(L \cap F) \), the events are not independent. This mathematical relationship signifies that there is some sort of interaction or dependency between the female respondents and those who learned from TV shows.
Survey Analysis
Survey analysis is a powerful tool in statistics for understanding populations based on sample data. When executing a survey, careful consideration is given to how the sample is selected鈥攁iming for a representative sample that mirrors the wider population.
In the provided exercise, a survey gathered responses from approximately 3500 people with the aim of representing adult Americans. By analyzing these survey results, statisticians obtain insights into various demographics, such as gender, to estimate behavioral trends in the broader population.

Design and Interpretation

The design of a survey impacts the reliability of its results. Proper sampling ensures that findings can be generalized to a larger population. Here, the task was to determine how certain demographics, such as gender, relate to learning about health issues through television.
Findings like those seen in the exercise offer decision-makers data-driven insights. For instance, the discovered dependencies between events \( L \) and \( F \) could guide media executives or health marketers on how to tailor their messages more effectively for different segments of the population. Understanding survey analysis enables better strategic planning and targeting to meet the needs and preferences of specific groups, ensuring communication and resources reach the intended audiences efficiently.

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Most popular questions from this chapter

A certain company sends \(40 \%\) of its overnight mail parcels by means of express mail service \(A_{1}\). Of these parcels, \(2 \%\) arrive after the guaranteed delivery time (use \(L\) to denote the event late delivery). If a record of an overnight mailing is randomly selected from the company's files, what is the probability that the parcel went by means of \(A_{1}\) and was late?

The accompanying probabilities are from the report "Estimated Probability of Competing in Athletics Beyond the High School Interscholastic Level" (www.ncaa.org). The probability that a randomly selected high school basketball player plays NCAA basketball as a freshman in college is \(.0303\); the probability that someone who plays NCAA basketball as a freshman will be playing NCAA basketball in his senior year is .7776; and the probability that a college senior NCAA basketball player will play professionally after college is .0102. Suppose that a high school senior basketball player is chosen at random. Define the events \(F, S\), and \(D\) as \(F=\) the event that the player plays as a college freshman \(S=\) the event that the player also plays as a senior \(D=\) the event that the player plays professionally after college a. Based on the information given, what are the values of \(P(F), P(S \mid F)\), and \(P(D \mid S \cap F)\) ? b. What is the probability that a senior high school basketball player plays college basketball as a freshman and as a senior and then plays professionally? (Hint: This is \(P(F \cap S \cap D) .)\)

Approximately \(30 \%\) of the calls to an airline reservation phone line result in a reservation being made. a. Suppose that an operator handles 10 calls. What is the probability that none of the 10 calls result in a reservation? b. What assumption did you make to calculate the probability in Part (a)? c. What is the probability that at least one call results in a reservation being made?

After all students have left the classroom, a statistics professor notices that four copies of the text were left under desks. At the beginning of the next lecture, the professor distributes the four books at random to the four students \((1,2,3\), and 4\()\) who claim to have left books. One possible outcome is that 1 receives 2's book, 2 receives 4's book, 3 receives his or her own book, and 4 receives l's book. This outcome can be abbreviated \((2,4,3,1)\). a. List the 23 other possible outcomes. b. Which outcomes are contained in the event that exactly two of the books are returned to their correct owners? Assuming equally likely outcomes, what is the probability of this event? c. What is the probability that exactly one of the four students receives his or her own book? d. What is the probability that exactly three receive their own books? e. What is the probability that at least two of the four students receive their own books?

The newspaper article "Folic Acid Might Reduce Risk of Down Syndrome" (USA Today, September 29 . 1999) makes the following statement: "Older women are at a greater risk of giving birth to a baby with Down Syndrome than are younger women. But younger women are more fertile, so most children with Down Syndrome are born to mothers under \(30 . "\) Let \(D=\) event that a randomly selected baby is born with Down Syndrome and \(Y=\) event that a randomly selected baby is born to a young mother (under age 30 ). For each of the following probability statements, indicate whether the statement is consistent with the quote from the article, and if not, explain why not. a. \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.004, P(Y)=.7\) b. \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.001, P(Y)=.7\) c. \(P(D \mid Y)=.004, P\left(D \mid Y^{C}\right)=.004, P(Y)=.7\) d. \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.004, P(Y)=.4\) e. \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.001, P(Y)=.4\) f. \(P(D \mid Y)=.004, P\left(D \mid Y^{C}\right)=.004, P(Y)=.4\)
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