5. Suppose a test for a disease is 85% accurate for those who have the disease (true positives) and 85% accurate for those who do not have the disease (true negatives). Within a sample of 4000 patients, the incidence rate of the disease is the national average. Complete parts (a) through (c) below. Disease No Disease Total Tested positive 51 591 642 Tested negative 9. 3349 3358 Total 60 3940 4000 a. Of those with the disease, what percentage test positive? b. Of those who test positive, what percentage have the disease? C. Compare the results in parts (a) and (b) and explain why they are different.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![### Understanding Disease Test Accuracy
#### Problem Statement
Suppose a test for a disease is 85% accurate for those who have the disease (true positives) and 85% accurate for those who do not have the disease (true negatives). Within a sample of 4000 patients, the incidence rate of the disease is the national average. Complete parts (a) through (c) below.
#### Data Table:
| | Tested Positive | Tested Negative | Total |
|-----------------------|-----------------|-----------------|-------|
| **Disease** | 51 | 9 | 60 |
| **No Disease** | 591 | 3349 | 3940 |
| **Total** | 642 | 3358 | 4000 |
### Questions:
a. **Of those with the disease, what percentage test positive?**
To find the percentage of people with the disease who test positive:
\[ \text{Percentage} = \left( \frac{\text{Number who tested positive and have the disease}}{\text{Total with the disease}} \right) \times 100 \]
So,
\[ \text{Percentage} = \left( \frac{51}{60} \right) \times 100 \approx 85\% \]
b. **Of those who test positive, what percentage have the disease?**
To find the percentage of people who test positive and actually have the disease:
\[ \text{Percentage} = \left( \frac{\text{Number who tested positive and have the disease}}{\text{Total who tested positive}} \right) \times 100 \]
So,
\[ \text{Percentage} = \left( \frac{51}{642} \right) \times 100 \approx 7.95\% \]
c. **Compare the results in parts (a) and (b) and explain why they are different.**
- **Part (a)** represents the sensitivity of the test, which is the probability that the test correctly identifies someone with the disease as positive. This is high at 85%.
- **Part (b)** represents the positive predictive value of the test, which is the probability that someone who tests positive actually has the disease. This is much lower at approximately 7.95%.
This discrepancy can be explained by the incidence rate of the disease being relatively low. Even though the test is highly](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d4eded-f6ef-4371-959f-6d41f62924d5%2Fd3992363-aed1-4a92-9af0-e7aca80d0594%2Fjo9x4ax_processed.jpeg&w=3840&q=75)
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