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Chapter 6: Problem 31

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. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.7\) b. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.001, \quad P(Y)=.7\) c. \(P(D \mid Y)=.004, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.7\) d. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.4\) e. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.001, \quad P(Y)=.4\) f. \(P(D \mid Y)=.004, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.4\)

Short Answer

Expert verified
Only statement a is consistent with the quote from the article. All other statements (b, c, d, e, f) are inconsistent because they either have equal risk of Down Syndrome for both young and old mothers, or the likelihood of a baby being born to a young mother is less than that of an old mother.

Step by step solution

01

Analyzing Statement a

The given statement shows \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.004, P(Y)=.7\). Here, the probability of Down Syndrome for babies born to younger mothers is less than the probability for babies born to older mothers, and the probability of a baby being born to a young mother is more than the baby being born to an older mother. Therefore, this is consistent with the quote.
02

Analyzing Statement b

The given statement shows \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.001, P(Y)=.7\). Here, the probability of Down Syndrome for babies born to younger mothers is the same as the probability for babies born to older mothers. This conflicts with the quote that states the risk is higher for older women, so it is inconsistent.
03

Analyzing Statement c

The given statement shows \(P(D \mid Y)=.004, P\left(D \mid Y^{C}\right)=.004, P(Y)=.7\). Here again, the probability of Down Syndrome for babies born to younger mothers is the same as for babies born to older mothers. Since it conflicts with the fact that the risk is higher for older women, it is inconsistent.
04

Analyzing Statement d

The given statement shows \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.004, P(Y)=.4\). Here, the probability of Down Syndrome for babies born to younger mothers is less than the probability for babies born to older mothers. However, the probability a baby being born to a young mother is less than the mother being older which contradicts the assumption in the article. Therefore, this is inconsistent.
05

Analyzing Statement e

The given statement shows \(P(D \mid Y)=.001, P\left(D \mid Y^{C}\right)=.001, P(Y)=.4\). In this case, the probabilities of Down Syndrome for babies born to younger and older mothers are the same which contradicts the article. Moreover, the total probability of a baby being born to a young mother is less than being born to an older mother. Thus, it is inconsistent.
06

Analyzing Statement f

The given statement shows \(P(D \mid Y)=.004, P\left(D \mid Y^{C}\right)=.004, P(Y)=.4\). Here, the probabilities of Down Syndrome for babies born to younger and older mothers are equal, and the total probability of a baby being born to a young mother is less than being born to an older mother. This statement is inconsistent with the article's assumptions.

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Key Concepts

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

Statistics
Statistics is the science of collecting, analyzing, presenting, and interpreting data. It helps us understand and describe the world around us. For example, in our exercise, statistics play a crucial role in interpreting data about Down Syndrome and its association with different age groups of mothers. With statistical analysis, one could determine the probability of certain outcomes, draw conclusions from data sets, and support claims with numerical evidence.

In the given problem, different probability values are used to evaluate statements concerning the likelihood of a baby having Down Syndrome, given the age of the mother. These values help assess the consistency of the newspaper article's claims. Statistics provides the framework to understand these probabilities and derive meaning from them, helping to communicate risks more effectively.
Probability Theory
Probability theory is the mathematical framework used to quantify uncertainty. It answers questions like "How likely is this event to happen?". In our problem scenario, it involves evaluating how likely it is for a baby to be born with Down Syndrome depending on whether the mother is younger or older than 30.

Probability theory includes concepts like conditional probability, which is used here to evaluate statements such as \(P(D \mid Y)\). Conditional probability considers the likelihood of event \(D\) given that \(Y\) has occurred. In our exercise, conditional probabilities compare the likelihood that a baby has Down Syndrome based on the age of the mother, and whether this aligns with the information given in the article.
Data Analysis
Data analysis is the process of inspecting, cleaning, transforming, and modeling data to discover useful information and support decision-making. It is essential in understanding patterns and trends in data. This is particularly relevant in our case study, where data concerning births and Down Syndrome is analyzed to verify claims made in the news article.

To perform data analysis effectively, it is essential to consider the quality and reliability of data. One must evaluate the consistency of data with assumptions and pre-existing information. In our exercise, the data analysis involves checking whether the assigned probabilities align with the assertion from the article, which states that younger mothers are statistically more likely to give birth to children with Down Syndrome despite a lower probability of Down Syndrome compared to older mothers. Such analysis helps authenticate or refute statements by applying statistical principles.

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Most popular questions from this chapter

The following case study was reported in the article "Parking Tickets and Missing Women," which appeared in an early edition of the book Statistics: A Guide to the \(U n\) known. In a Swedish trial on a charge of overtime parking, a police officer testified that he had noted the position of the two air valves on the tires of a parked car: To the closest hour, one was at the one o'clock position and the other was at the six o'clock position. After the allowable time for parking in that zone had passed, the policeman returned, noted that the valves were in the same position, and ticketed the car. The owner of the car claimed that he had left the parking place in time and had returned later. The valves just happened by chance to be in the same positions. An "expert" witness computed the probability of this occurring as \((1 / 12)(1 / 12)=1 / 44\). a. What reasoning did the expert use to arrive at the probability of \(1 / 44\) ? b. Can you spot the error in the reasoning that leads to the stated probability of \(1 / 44\) ? What effect does this error have on the probability of occurrence? Do you think that \(1 / 44\) is larger or smaller than the correct probability of occurrence?

The student council for a school of science and math has one representative from each of the five academic departments: biology (B), chemistry (C), mathematics (M), physics (P), and statistics (S). Two of these students are to be randomly selected for inclusion on a university-wide student committee (by placing five slips of paper in a bowl, mixing, and drawing out two of them). a. What are the 10 possible outcomes (simple events)? b. From the description of the selection process, all outcomes are equally likely; what is the probability of each simple event? c. What is the probability that one of the committee members is the statistics department representative? d. What is the probability that both committee members come from laboratory science departments?

Information from a poll of registered voters in Cedar Rapids, Iowa, to assess voter support for a new school tax was the basis for the following statements (Cedar Rapids Gazette, August 28,1999 ): The poll showed 51 percent of the respondents in the Cedar Rapids school district are in favor of the tax. The approval rating rises to 56 percent for those with children in public schools. It falls to 45 percent for those with no children in public schools. The older the respondent, the less favorable the view of the proposed tax: 36 percent of those over age 56 said they would vote for the tax compared with 72 percent of 18- to 25 -year-olds. Suppose that a registered voter from Cedar Rapids is selected at random, and define the following events: \(F=\) event that the selected individual favors the school \(\operatorname{tax}, C=\) event that the selected individual has children in the public schools, \(O=\) event that the selected individual is over 56 years old, and \(Y=\) event that the selected individual is \(18-25\) years old. a. Use the given information to estimate the values of the following probabilities: i. \(P(F)\) ii. \(P(F \mid C)\) iii. \(P\left(F \mid C^{C}\right)\) iv. \(P(F \mid O)\) v. \(P(F \mid Y)\) b. Are \(F\) and \(C\) independent? Justify your answer. c. Are \(F\) and \(O\) independent? Justify your answer.

Insurance status - covered (C) or not covered (N) \- is determined for each individual arriving for treatment at a hospital's emergency room. Consider the chance experiment in which this determination is made for two randomly selected patients. The simple events are \(O_{1}=(\mathrm{C}, \mathrm{C})\) \(O_{2}=(\mathrm{C}, \mathrm{N}), O_{3}=(\mathrm{N}, \mathrm{C})\), and \(O_{4}=(\mathrm{N}, \mathrm{N}) .\) Suppose that probabilities are \(P\left(O_{1}\right)=.81, P\left(O_{2}\right)=.09, P\left(O_{3}\right)=.09\), and \(P\left(O_{4}\right)=.01\). a. What outcomes are contained in \(A\), the event that at most one patient is covered, and what is \(P(A)\) ? b. What outcomes are contained in \(B\), the event that the two patients have the same status with respect to coverage, and what is \(P(B)\) ?

In an article that appears on the web site of the American Statistical Association (www.amstat.org), Carlton Gunn, a public defender in Seattle, Washington, wrote about how he uses statistics in his work as an attorney. He states: I personally have used statistics in trying to challenge the reliability of drug testing results. Suppose the chance of a mistake in the taking and processing of a urine sample for a drug test is just 1 in 100 . And your client has a "dirty" (i.e., positive) test result. Only a 1 in 100 chance that it could be wrong? Not necessarily. If the vast majority of all tests given - say 99 in 100 \- are truly clean, then you get one false dirty and one true dirty in every 100 tests, so that half of the dirty tests are false. Define the following events as \(T D=\) event that the test result is dirty, \(T C=\) event that the test result is clean, \(D=\) event that the person tested is actually dirty, and \(C=\) event that the person tested is actually clean. a. Using the information in the quote, what are the values of \mathbf{i} . ~ \(P(T D \mid D)\) iii. \(P(C)\) ii. \(P(T D \mid C) \quad\) iv. \(P(D)\) b. Use the law of total probability to find \(P(T D)\). c. Use Bayes' rule to evaluate \(P(C \mid T D)\). Is this value consistent with the argument given in the quote? Explain.
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