If a test for a disease has a sensitivity of 90% and a specificity of 95%, what
Practice Questions
Q1
If a test for a disease has a sensitivity of 90% and a specificity of 95%, what is the probability that a person who tests positive actually has the disease, given that the prevalence is 1%? (2022)
0.018
0.1
0.5
0.9
Questions & Step-by-Step Solutions
If a test for a disease has a sensitivity of 90% and a specificity of 95%, what is the probability that a person who tests positive actually has the disease, given that the prevalence is 1%? (2022)
Step 1: Understand the terms. Sensitivity is the probability that the test correctly identifies a person with the disease. Specificity is the probability that the test correctly identifies a person without the disease. Prevalence is the actual proportion of people in the population who have the disease.
Step 2: Write down the values given in the question. Sensitivity = 90% = 0.9, Specificity = 95% = 0.95, Prevalence = 1% = 0.01.
Step 3: Calculate the probability of having the disease given a positive test result using Bayes' theorem. The formula is: P(Disease | Positive) = (Sensitivity × Prevalence) / ((Sensitivity × Prevalence) + (1 - Specificity) × (1 - Prevalence)).
Step 4: Substitute the values into the formula. P(Disease | Positive) = (0.9 × 0.01) / ((0.9 × 0.01) + (0.05 × 0.99)).
Step 5: Calculate the numerator: 0.9 × 0.01 = 0.009.
Step 7: Divide the numerator by the denominator: 0.009 / 0.0585 = 0.1538.
Step 8: Convert the result to a percentage: 0.1538 × 100 = 15.38%.
Step 9: The final answer is that the probability that a person who tests positive actually has the disease is approximately 15.38%.
Bayes' Theorem – A mathematical formula used to calculate the probability of a condition based on prior knowledge and new evidence.
Sensitivity and Specificity – Sensitivity measures the proportion of actual positives correctly identified, while specificity measures the proportion of actual negatives correctly identified.
Prevalence – The proportion of a population found to have a condition at a specific time.
