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Chapter 5: Problem 31

The Associated Press (San Luis Obispo Telegram-Tribune, August 23,1995 ) reported the results of a study in which schoolchildren were screened for tuberculosis (TB). It was reported that for Santa Clara County, California, the proportion of all tested kindergartners who were found to have TB was 0.0006 . The corresponding proportion for recent immigrants (thought to be a high-risk group) was \(0.0075 .\) Suppose that a Santa Clara County kindergartner is to be selected at random. Are the events selected student is a recent immigrant and selected student has \(T B\) independent or dependent events? Justify your answer using the given information.

Short Answer

Expert verified
Due to the lack of specific information needed, it's inconclusive whether the events 'selected student is a recent immigrant' and 'selected student has TB' are independent or dependent events.

Step by step solution

01

Understand Probabilities

First, understand that a probability can be equated to the given proportions. Using the information provided, the probability that a randomly chosen kindergartner has TB, denoted as P(TB), is 0.0006. The probability that a randomly selected kindergartner is a recent immigrant and has TB, denoted as P(Immigrant and TB), is 0.0075.
02

Understand Independence

Events are said to be independent if the probability of both occurring is the product of the probabilities of each occurring. In other words, P(A and B) = P(A) * P(B) for independent events A and B. Here, we are asked to determine if the events 'selected student is a recent immigrant' and 'selected student has TB' are independent.
03

Apply Rule of Independence

If the events were independent, then the probability that a kindergarten student is a recent immigrant and has TB would be the product of the probabilities of each event. Based on our understanding of how the probabilities are set up in this situation, there is not enough information provided to calculate the probability that a kindergartner is a recent immigrant, required for the independence check. Therefore, we cannot say for certain whether these events are independent or dependent.
04

Conclude answer

With the provided information, we cannot conclusively determine whether the events 'selected student is a recent immigrant' and 'selected student has TB' are independent or dependent events. The specific probability of a kindergartner being a recent immigrant is needed to accomplish this.

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Most popular questions from this chapter

Suppose you want to estimate the probability that a patient will develop an infection while hospitalized at a particular hospital. In the past year, this hospital had 6,450 patients, and 712 of them developed an infection. What is the estimated probability that a patient at this hospital will develop an infection?

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