https://file.aiquickdraw.com/mathbot/file/17289305165023peoi8kj.jpg

Let's go through the probability and statistics questions based on the table and problem presented in the image.

### Table Overview:
The table shows data for patients tested for a disease. It summarizes the results of the test:

|                  | Does not have disease | Has disease | Total |
|------------------|-----------------------|-------------|-------|
| **Negative**      | 31                    | 84          | 115   |
| **Positive**      | 14                    | 25          | 39    |
| **Total**         | 45                    | 109         | 154   |

Now, let's address each question one by one:

### a) Suppose a patient is chosen at random. What is the probability that the chosen patient has the disease and received a positive test result?

This is asking for the **joint probability** of having the disease and testing positive. 

From the table, 25 patients have the disease and tested positive. The total number of patients is 154.

\[
P(\text{Disease and Positive}) = \frac{25}{154}
\]

So, the probability is:
\[
P = 0.1623 \, (\text{or} \, 16.23\%)
\]

---

### b) Suppose a patient is chosen at random. What is the probability that the chosen patient has the disease?

This is asking for the **marginal probability** of having the disease.

From the table, 109 patients have the disease. The total number of patients is 154.

\[
P(\text{Disease}) = \frac{109}{154}
\]

So, the probability is:
\[
P = 0.7078 \, (\text{or} \, 70.78\%)
\]

---

### c) What is the sensitivity of the test?

**Sensitivity** (True Positive Rate) measures the proportion of actual positives (people with the disease) correctly identified by the test.

\[
\text{Sensitivity} = \frac{\text{True Positives}}{\text{Total Diseased}} = \frac{25}{109}
\]

So, the sensitivity is:
\[
= 0.2294 \, (\text{or} \, 22.94\%)
\]

---

### d) What is the specificity of the test?

**Specificity** (True Negative Rate) measures the proportion of actual negatives (people without the disease) correctly identified by the test.

\[
\text{Specificity} = \frac{\text{True Negatives}}{\text{Total Healthy}} = \frac{31}{45}
\]

So, the specificity is:
\[
= 0.6889 \, (\text{or} \, 68.89\%)
\]

---

### e) What is the false positive rate of the test?

The **False Positive Rate** is the proportion of people without the disease who incorrectly received a positive test result.

\[
\text{False Positive Rate} = \frac{\text{False Positives}}{\text{Total Healthy}} = \frac{14}{45}
\]

So, the false positive rate is:
\[
= 0.3111 \, (\text{or} \, 31.11\%)
\]

---

Would you like to go over the remaining questions (f-h) as well?

### Here are 5 additional related questions:
1. What is the false negative rate of the test?
2. How do you calculate the Positive Predictive Value (PPV) of the test?
3. What would happen to sensitivity if the test had fewer false negatives?
4. What is the impact of increasing specificity on the false positive rate?
5. How does the prevalence of a disease affect predictive values?

### Tip:
In medical tests, **sensitivity** and **specificity** are important for measuring the accuracy of the test. Sensitivity focuses on detecting true cases, while specificity focuses on ruling out healthy individuals.

A total of 154 patients have participated in a study to examine how well a new medical test can detect a disease. Each patient either has or does not have the disease, and the test can return a positive or negative result. The table summarizes the data from the study. The questions are related to probabilities, sensitivity, specificity, and other statistics based on this table.

Explore a study with 154 patients to analyze the probability, sensitivity, specificity, and false positive rate of a medical test designed to detect a disease. Learn how to calculate these important statistical values step-by-step using data from a test involving true positives, true negatives, and false positives.

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