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Chapter 3: Problem 1

In a study of the relationship between health risk and income, a large group of people living in Massachusetts were asked a series of questions. \({ }^{7}\) Some of the results are shown in the following table. $$ \begin{array}{|l|rrrr|} \hline && {\text { Income }} & \\ & \text { Low } & \text { Medium } & \text { High } & \text { Total } \\ \hline \text { Smoke } & 634 & 332 & 247 & 1,213 \\ \text { Don't smoke } & 1,846 & 1,622 & 1,868 & 5,336 \\ \text { Total } & 2,480 & 1,954 & 2,115 & 6,549 \\ \hline \end{array} $$ (a) What is the probability that someone in this study smokes? (b) What is the conditional probability that someone in this study smokes, given that the person has high income? (c) Is being a smoker independent of having a high income? Why or why not?

Short Answer

Expert verified
The probability of smoking is 0.185, the conditional probability of smoking given high income is 0.1168, and smoking is not independent of having a high income because the conditional probability is not equal to the overall probability of smoking.

Step by step solution

01

Calculate the Probability of Smoking

To find the probability that someone in the study smokes, divide the total number of people who smoke by the total number of people in the study. The number of people who smoke is 1,213 and the total number of people in the study is 6,549.
02

Calculate the Conditional Probability of Smoking Given High Income

To calculate the conditional probability, divide the number of high-income people who smoke by the total number of high-income people. The number of high-income smokers is 247 and the total number of high-income people is 2,115.
03

Determine Independence

Two events A and B are independent if P(A | B) = P(A). Compare the probability of smoking with the probability of smoking given a high income to determine independence.

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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. It is represented by a number between 0 and 1, where 0 indicates an impossibility, and 1 indicates certainty. Calculating the probability of an event involves dividing the number of favorable outcomes by the total number of possible outcomes.

For example, in the context of the health risk and income study, to conclude the probability that someone smokes, we look at the total number of smokers (1,213) and the overall sample size (6,549). The probability (\( P \text{(Smoke)} \) is calculated using the formula: \[ P(\text{Smoke}) = \frac{\text{Number of Smokers}}{\text{Total Number of Participants}} = \frac{1,213}{6,549} \]. This result can then be interpreted as the likelihood of picking someone who smokes from the whole group of participants.

Understanding how to calculate simple probabilities is crucial because it forms the basis for more advanced topics, such as conditional probability and statistical independence.
Health Risk and Income Study
A health risk and income study typically aims to understand if there is any correlation between health-related behaviors, like smoking, and income levels. Studies like the one described in our exercise categorize participants into different income brackets—low, medium, and high—and record their habits or conditions.

Conducting this kind of research requires careful collection of data and subsequent analysis to draw meaningful conclusions. The table in the exercise presents a clear overview of the participants' smoking status across various income levels. Such studies can reveal important public health patterns, influencing policy-making and targeted interventions.

Improving the quality of such studies involves ensuring that a representative sample of the population is included, that the data collection methods are reliable, and that the categorizations of income levels reflect meaningful differences in economic status.
Statistical Independence
Statistical independence is a key concept in probability that describes a scenario where the occurrence (or non-occurrence) of one event does not affect the probability of another event. Two events are independent if the probability of one event occurring is the same whether or not the other event occurs.

In the context of the exercise, to explore whether the event of 'being a smoker' is independent of 'having a high income,' we compare the general probability of smoking to the conditional probability of smoking given high income. If the two probabilities are equal, the events are independent. Formally, if \( P(\text{Smoke}) = P(\text{Smoke|High Income}) \), the two events are independent.

To find this out from our given data, if the probability of smoking is the same regardless of income level, there is independence between smoking and income level. However, if the probabilities differ, this suggests a relationship between the two, and thus, they are not statistically independent.

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Most popular questions from this chapter

The seeds of the garden pea (Pisum sativum) are either yellow or green. A certain cross between pea plants produces progeny in the ratio 3 yellow : 1 green. \({ }^{16}\) If four randomly chosen progeny of such a cross are examined, what is the probability that (a) three are yellow and one is green? (b) all four are yellow? (c) all four are the same color?

An important method for studying mutation-causing substances involves killing female mice 17 days after mat ing and examining their uteri for living and dead embryos. The classical method of analysis of such data assumes that the survival or death of each embryo constitutes an independent binomial trial. The accompanying table, which is extracted from a larger study, gives data for 310 females, all of whose uteri contained 9 embryos; all of the animals were treated alike (as controls). \(^{25}\) (a) Fit a binomial distribution to the observed data. (Round the expected frequencies to one decimal place.) (b) Interpret the relationship between the observed and expected frequencies. Do the data cast suspicion on the classical assumption? $$ \begin{array}{|ccc|} \hline \text { Number of embryos } & \text { Number of } \\ \hline \text { Dead } & \text { Living } & \text { female mice } \\ \hline 0 & 9 & 136 \\ 1 & 8 & 103 \\ 2 & 7 & 50 \\ 3 & 6 & 13 \\ 4 & 5 & 6 \\ 5 & 4 & 1 \\ 6 & 3 & 1 \\ 7 & 2 & 0 \\ 8 & 1 & 0 \\ 9 & 0 & 0 \\ \hline \end{array} $$

Neuroblastoma is a rare, serious, but treatable disease. A urine test, the VMA test, has been developed that gives a positive diagnosis in about \(70 \%\) of cases of neuroblastoma. \({ }^{22}\) It has been proposed that this test be used for large-scale screening of children. Assume that 300,000 children are to be tested, of whom 8 have the disease. We are interested in whether or not the test detects the disease in the 8 children who have the disease. Find the probability that (a) all eight cases will be detected. (b) only one case will be missed. (c) two or more cases will be missed. [Hint: Use parts (a) and (b) to answer part (c).]

In a certain population of the European starling, there are 5,000 nests with young. The distribution of brood size (number of young in a nest) is given in the accompanying table. \(^{12}\) $$ \begin{array}{|cc|} \hline \text { Brood Size } & \text { Frequency (No. of Broods) } \\ \hline 1 & 90 \\ 2 & 230 \\ 3 & 610 \\ 4 & 1,400 \\ 5 & 1,760 \\ 6 & 750 \\ 7 & 130 \\ 8 & 26 \\ 9 & 3 \\ 10 & 1 \\ \hline \text { Total } & 5,000 \\ \hline \end{array} $$ Suppose one of the 5,000 broods is to be chosen at random, and let \(Y\) be the size of the chosen brood. Find (a) \(\operatorname{Pr}\\{Y=3\\}\) (b) \(\operatorname{Pr}\\{Y \geq 7\\}\) (c) \(\operatorname{Pr}\\{4 \leq Y \leq 6\\}\)

The sex ratio of newborn human infants is about 105 males : 100 females. \(^{21}\) If four infants are chosen at random, what is the probability that (a) two are male and two are female? (b) all four are male? (c) all four are the same sex?
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