# Analysis of Covid-19 Test Results ## Key Points: - The test's **accuracy** varies depending on the true infection status. - The **probability** that a person is infected given a positive test is called the **positive predictive value**. - The **prob** that a person is not infected given a negative test is called the **negative predictive value**. --- ## Given Data: | | Test Positive | Test Negative | Total | |-------------------|-----------------|-----------------|---------| | Infected | .01 | .2 | .5 | | Not Infected | .001 | .5 | | | **Total** | | | 1. | --- ## (a) Probability the person is infected given a positive test ### Calculation: \[ P(\text{Infected} | \text{Test Positive}) = \frac{P(\text{Test Positive} | \text{Infected}) \times P(\text{Infected})}{P(\text{Test Positive})} \] Where: \[ P(\text{Test Positive}) = P(\text{Test Positive} | \text{Infected}) \times P(\text{Infected}) + P(\text{Test Positive} | \text{Not Infected}) \times P(\text{Not Infected}) \] Plugging in the values: \[ P(\text{Test Positive}) = .01 \times .5 + .001 \times .5 = .005 + .0005 = .0055 \] Now: \[ P(\text{Infected} | \text{Test Positive}) = \frac{.01 \times .5}{.0055} = \frac{.005}{.0055} \approx .909 \] **Answer:** The probability the person is infected given a positive test is approximately **90.9%**. --- ## (b) Probability the test yields a positive result in this community ### Calculation: \[ P(\text{Test Positive}) = .0055 \] **Answer:** The probability the test yields a positive result is **.55%**. --- ## Summary: - **Probability infected given positive test:** **~90.9%** - **Probability positive test in the community:** **.55%**