4. Probability
Addition Rule
4:28 minutesProblem 4.4.21c
Textbook Question
In Exercises 21–24, use these results from the “1-Panel-THC” test for marijuana use, which is provided by the company Drug Test Success: Among 143 subjects with positive test results, there are 24 false positive (incorrect) results; among 157 negative results, there are 3 false negative (incorrect) results. (Hint: Construct a table similar to Table 4-1.)
Testing for Marijuana Use
c. What is the probability that a randomly selected subject had a true negative result?
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the probability that a randomly selected subject had a true negative result. A true negative result occurs when the test correctly identifies a subject as not using marijuana.
Step 2: Organize the data into a contingency table. From the problem, we know: (1) There are 143 positive test results, of which 24 are false positives. This means the number of true positives is 143 - 24 = 119. (2) There are 157 negative test results, of which 3 are false negatives. This means the number of true negatives is 157 - 3 = 154.
Step 3: Calculate the total number of subjects. Add the total positive and negative test results: 143 (positive) + 157 (negative) = 300 subjects.
Step 4: Define the probability formula for a true negative result. The probability of a true negative is given by the formula: \( P(\text{True Negative}) = \frac{\text{Number of True Negatives}}{\text{Total Number of Subjects}} \).
Step 5: Substitute the values into the formula. Use the number of true negatives (154) and the total number of subjects (300) to compute the probability. The result will be \( P(\text{True Negative}) = \frac{154}{300} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
True Negative Rate
The true negative rate, also known as specificity, measures the proportion of actual negatives that are correctly identified by a test. In this context, it refers to the number of subjects who tested negative for marijuana use and were indeed not using it, divided by the total number of subjects who were not using marijuana. Understanding this concept is crucial for calculating the probability of a true negative result.
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Probability Calculation
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. To find the probability of a true negative result, one must use the formula: P(True Negative) = Number of True Negatives / Total Number of Subjects. This calculation helps quantify the effectiveness of the test in identifying non-users.
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Confusion Matrix
A confusion matrix is a table used to evaluate the performance of a classification model by displaying the true positives, true negatives, false positives, and false negatives. In this scenario, constructing a confusion matrix will help visualize the test results and facilitate the calculation of probabilities related to true and false results, providing a clearer understanding of the test's accuracy.
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