A certain virus affects 0.7% of the population. A test used to detect the virus in a person is positive 87% of the time if the person has the virus (true positive) and 14% of the time if the person does not have the virus (false positive). Fill out the remainder of the following table and use it to answer the two questions below based on a total sample of 100,000 people.

Virus

No Virus

Total

Positive Test

Negative Test

Total

100,000

To solve this problem, we need to fill out the table using the given probabilities and then calculate the requested values.

**Step 1: Define the given probabilities and calculations.**

1. **Probability of having the virus**: 0.7% (or 0.007).
2. **Probability of not having the virus**: 99.3% (or 0.993).
3. **True positive rate** (test positive if person has the virus): 87% (or 0.87).
4. **False positive rate** (test positive if person does not have the virus): 14% (or 0.14).

**Step 2: Calculate the expected numbers for each category based on a sample of 100,000 people.**

1. **Number of people with the virus**:
   \[
   \text{People with virus} = 100,000 \times 0.007 = 700
   \]

2. **Number of people without the virus**:
   \[
   \text{People without virus} = 100,000 \times 0.993 = 99,300
   \]

**Step 3: Determine the number of positive and negative test results.**

**For people with the virus:**
- **True Positives (Positive Test with Virus)**:
  \[
  \text{True Positives} = 700 \times 0.87 = 609
  \]
- **False Negatives (Negative Test with Virus)**:
  \[
  \text{False Negatives} = 700 - 609 = 91
  \]

**For people without the virus:**
- **False Positives (Positive Test without Virus)**:
  \[
  \text{False Positives} = 99,300 \times 0.14 = 13,902
  \]
- **True Negatives (Negative Test without Virus)**:
  \[
  \text{True Negatives} = 99,300 - 13,902 = 85,398
  \]

**Step 4: Fill in the table.**

\[
\begin{array}{|c|c|c|c|}
\hline
& \textbf{Virus} & \textbf{No Virus} & \textbf{Total} \\
\hline
\textbf{Positive Test} & 609 & 13,902 & 14,511 \\
\hline
\textbf{Negative Test} & 91 & 85,398 & 85,489 \\
\hline
\textbf{Total} & 700 & 99,300 & 100,000 \\
\hline
\end{array}
\]

**Step 5: Answer the questions based on the table.**

1. **Probability that a person who tests positive actually has the virus (Positive Predictive Value)**:
   \[
   P(\text{Virus} | \text{Positive Test}) = \frac{\text{True Positives}}{\text{Total Positive Tests}} = \frac{609}{14,511} \approx 0.042 \text{ or } 4.2\%
   \]

2. **Probability that a person who tests negative does not have the virus (Negative Predictive Value)**:
   \[
   P(\text{No Virus} | \text{Negative Test}) = \frac{\text{True Negatives}}{\text{Total Negative Tests}} = \frac{85,398}{85,489} \approx 0.999 \text{ or } 99.9\%
   \]

Would you like more details or have any questions?

**Relative Questions:**
1. How would the values change if the prevalence of the virus increased?
2. What is the impact of increasing the test's true positive rate on the predictive values?
3. How does the false positive rate affect the reliability of the test?
4. How would a change in sample size affect the results?
5. How could improving test accuracy affect public health decisions?

**Tip:** The Positive Predictive Value is highly influenced by the prevalence of the condition in the population; lower prevalence leads to lower predictive values even if the test is highly accurate.

Explore a detailed analysis of virus detection test results using probability and statistics. Understand how to interpret true positive, false positive, true negative, and false negative rates. Includes calculations based on a sample of 100,000 people.

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