4. Probability
Multiplication Rule: Independent Events
1:47 minutesProblem 4.2.21a
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
a. How many subjects are included in the study?
Verified step by step guidance1
Step 1: Understand the problem. The problem involves analyzing the results of a drug test for marijuana use. We are tasked with determining the total number of subjects included in the study. The key information provided includes the number of positive and negative test results, as well as the number of false positives and false negatives.
Step 2: Identify the components of the study. There are 143 subjects with positive test results, which include 24 false positives. There are also 157 subjects with negative test results, which include 3 false negatives.
Step 3: Calculate the total number of subjects. The total number of subjects is the sum of all positive test results and all negative test results. Use the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>Total</mi><mo>=</mo><mi>Positive</mi><mo>+</mo><mi>Negative</mi></math>.
Step 4: Substitute the given values into the formula. Replace 'Positive' with 143 and 'Negative' with 157 in the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>Total</mi><mo>=</mo><mn>143</mn><mo>+</mo><mn>157</mn></math>.
Step 5: Conclude the calculation. Add the two values together to find the total number of subjects in the study. This will give you the final answer.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
False Positives and False Negatives
False positives occur when a test incorrectly indicates the presence of a condition, while false negatives occur when a test fails to detect a condition that is present. In the context of drug testing, understanding these terms is crucial for interpreting the accuracy of the test results and assessing the reliability of the testing method.
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Correlation Coefficient
Total Subjects in a Study
The total number of subjects in a study is the sum of all individuals who participated, regardless of their test results. In this case, it includes both those who tested positive and negative for marijuana use, which is essential for calculating overall statistics and understanding the study's scope.
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Contingency Table
A contingency table is a data representation that displays the frequency distribution of variables, allowing for easy comparison of outcomes. In this scenario, constructing a contingency table will help visualize the relationships between test results (positive/negative) and their accuracy (true/false), facilitating a clearer analysis of the data.
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