Use these results from the “1PanelTHC” test for marijuana...

Use these results from the “1PanelTHC” 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 results; among 157 negative results, there are 3 false negative results. (Hint: Construct a table similar to Table 41, which is included with the Chapter Problem.) If 3 of the subjects are randomly selected without replacement, what is the probability that they all had true negative test results? Is such an event unlikely?

Use these results from the “1PanelTHC” 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 results; among 157 negative results, there are 3 false negative
results. (Hint: Construct a table similar to Table 41, which is included with the Chapter Problem.)
If 3 of the subjects are randomly selected without replacement, what is the probability that they all had true negative test results? Is such an event unlikely?

Key Concepts

Contingency Table Construction

A contingency table is a matrix used to display the frequency distribution of variables, often used to summarize the performance of diagnostic tests by categorizing outcomes into true positive, false positive, false negative, and true negative. This organization aids in understanding the relationships between test results and actual conditions.

Diagnostic Test Accuracy Metrics

Diagnostic test accuracy metrics include concepts such as sensitivity, specificity, positive predictive value, and negative predictive value. These metrics are derived from the counts in a contingency table and help in assessing the effectiveness and reliability of a test in identifying the presence or absence of a condition.

False Positives and False Negatives

False positives and false negatives are types of errors in diagnostic testing. A false positive occurs when a test incorrectly indicates the presence of a condition, while a false negative occurs when a test fails to detect a condition that is present. Recognizing these errors is fundamental to interpreting the accuracy of a test.

Probability without Replacement

Probability without replacement refers to the method of calculating events where each selection is drawn from a finite population and is not returned, causing the probabilities to change with each draw. This concept is crucial when considering scenarios where multiple individuals are selected sequentially, as each selection affects the subsequent probability.

Hypergeometric Distribution

The hypergeometric distribution is a probability distribution used for sampling without replacement from a finite population. It is particularly relevant when determining the probability of a predefined number of successes (such as true negatives) in a sample, and it accurately models scenarios where the outcomes are dependent on each other.

Interpreting Unlikely Events

Interpreting an event as 'unlikely' involves comparing its calculated probability to a benchmark or expected probability. In statistical terms, if an event's probability is very low, it may be considered an outlier or a rare occurrence, which can be significant in contexts such as reliability testing and quality control.

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