Question

The Associated Press (San Luis Obispo TelegramTribune, August 23,1995 ) reported on the results of mass screening of schoolchildren for tuberculosis (TB). For Santa Clara County, California, the proportion of all tested kindergartners who were found to have TB was .0006. The corresponding proportion for recent immigrants (thought to be a high-risk group) was .0075. Suppose that a Santa Clara County kindergartner is 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

With the given data, it's impossible to definitively determine if the events 'selected student is a recent immigrant' and 'selected student has TB' are independent or dependent.

Step-by-step solution

Formulate the given data

We have two events here:\nEvent A: Selected student is a recent immigrant.\nEvent B: Selected student has TB.\nThe proportion of all tested kindergartners who were found to have TB is 0.0006, we denote it as P(B).\nThe corresponding proportion for recent immigrants (thought to be a high-risk group) was 0.0075, we represent it as P(A and B). Let's assume P(A) denotes the probability that a randomly selected kindergartener is a recent immigrant. Unfortunately, we do not have this data point explicitly stated. Therefore, without this data point, we can't definitively classify the events as independent or dependent.

Attempt to classify the event based on provided data

We can tell that if P(A and B) = P(A) * P(B), then the events A and B are independent. However, we do not have the value of P(A), so we can't make this calculation. Additionally, we can't determine whether P(A|B) = P(A), i.e., the probability of event A given event B has occurred equals the probability of event A, because we also lack the values, which are necessary to calculate.

Concluding the analysis

Since we don't have all the necessary data points (in this case, the proportion of kindergartners who are recent immigrants or P(A)), we can't definitively determine whether the events are independent or dependent. With the given data, we simply can't make a justified decision.