91Ó°ÊÓ

In an article that appears on the website 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 \(C=\) event that the person tested is actually clean a. Using the information in the quote, what are the values of i. \(P(T D \mid D)\) iii. \(P(C)\) ii. \(P(T D \mid C)\) iv. \(P(D)\) b. Use the probabilities from Part (a) to construct a "hypothetical 1000 " table. c. What is the value of \(P(T D)\) ? d. Use the table to calculate the probability that a person is clean given that the test result is dirty, \(P(C \mid T D)\). Is this value consistent with the argument given in the quote? Explain.

Step by step solution

01

– Calculate Probabilities

Given information can be translated into probabilities as follows: \n The probability of the test result being dirty when the person tested is dirty, that is, \(P(T D \mid D) = 1\) or 100%.\n The probability of a test result being dirty when the person tested is actually clean, that is, \(P(T D \mid C) = 1/100\) or 1%.\n The probability of a person being tested is actually clean, which is, \(P(C) = 99/100\) or 99%.\n And finally, the probability of a person being tested is dirty, which is, \(P(D) = 1 – P(C) = 1/100\) or 1%.

02

– Construct a Hypothetical 1000 Table

With 1000 people tested, 990 would be clean and 10 would be dirty. Out of 990 clean, 99% or 981 would test correctly clean and 1% or 9 would test falsely dirty. For 10 dirty, all would test correctly dirty as \(P(T D | D) = 1\). So, we get a total of 19 dirty tests (9 false and 10 true).

03

– Calculate \(P(T D)\)

\(P(T D)\) would be the total number of dirty tests divided by the total number of tests. Here, we divide 19 (total) by 1000 (total tests) to get \(P(T D) = 19/1000\) or 1.9%.

04

– Calculate \(P(C \mid T D)\) and Analyze Its Consistency with Given Argument

Using Bayes’ theorem, we calculate \(P(C \mid T D)\) = \(P(T D \mid C)*P(C)/P(T D)\) which comes out to be \(0.474\), or 47.4%. As per the argument, with majority of tests being clean, half the dirty tests could be false. This aligns with our calculation where around 47.4%, nearly half of the dirty tests, could wrongly indicate a clean person as dirty.

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Key Concepts

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

False Positives

In statistics and testing scenarios, a false positive is when a test result incorrectly indicates the presence of a condition when it's not actually there. Imagine if you had a medical test that says you're sick, but you're actually healthy. That's a false positive.
A false positive rate is important because it tells how often we can expect errors in testing. In our exercise, the false positive rate was 1%, meaning out of 100 clean people, 1 could get a positive result falsely. This can have significant consequences.
Understanding false positives helps in deciding how reliable a test is. If a test frequently gives false positives, it may cause unnecessary stress or treatment for people who aren't truly affected. Always consider the false positive rate when evaluating test results to avoid misinterpretation.

Conditional Probability

Conditional probability helps us understand how likely something is to happen, given that something else has already happened. It's like saying, "If it rains tomorrow, what are the chances I'll stay indoors?". This is useful in our exercise where we are asked to evaluate outcomes based on certain conditions.
In mathematical terms, the probability of event A happening given event B has happened is written as \( P(A \mid B) \).
For our exercise:

• \( P(T D \mid D) = 1 \): The probability that a test is dirty given that the person is actually dirty.
• \( P(T D \mid C) = 0.01 \): The probability that a test is dirty given the person is actually clean.

These probabilities show the reliability of the results contingent upon certain conditions. Understanding these can guide more accurate interpretations of test results.

Hypothetical Reasoning

Hypothetical reasoning involves creating a scenario to understand potential outcomes. In our exercise, this involves using a 'hypothetical 1000 table'.
This table supposes 1000 people are tested. It helps illustrate probabilities by showing actual counts of outcomes based on given probabilities:

• 990 people are actually clean.
• 10 people are actually dirty.
• Out of the 990 clean individuals, 9 get incorrect dirty results (false positives).
• Out of the 10 truly dirty individuals, all receive dirty results.

By imagining this scenario, we can clearly see how many people are likely to fall into each category, including false positives. This method makes probabilistic concepts easier to comprehend by visualizing them in tangible terms. Such reasoning is crucial to validate claims and improve decision-making processes.