I need help with a and b. ### Understanding Diagnostic Test Performance in Detecting HypertensionA new diagnostic test is developed for detecting hypertension. This test is analyzed in a population where 22% of individuals have hypertension. The sensitivity (true positive rate) of the test is 85%, and its specificity (true negative rate) is 91%. Given these conditions, consider a random subject from this population. Here are some common probabilities one might want to calculate with the provided data:#### Task Analysisa. **Probability that the subject has hypertension and tests positive:** - This is the joint probability of having hypertension and a positive test result. It can be calculated using the sensitivity of the test and the prevalence of hypertension in the population.b. **Probability that the subject does not have hypertension and tests negative:** - This involves calculating the joint probability of not having hypertension and receiving a negative test result, using the specificity of the test and the complement of the prevalence.c. **Probability that the subject has hypertension given the subject tests positive:** - This is a conditional probability problem. It involves Bayes’ theorem to determine the likelihood of having hypertension given a positive test result.d. **Probability that the subject does not have hypertension given the subject tests negative:** - This scenario also uses conditional probability to find the likelihood of not having hypertension given a negative test result.e. **Probability that the subject does not have hypertension given the subject tests positive:** - This examines the opposite condition of part (c), figuring out the likelihood of being healthy despite a positive test result.### Calculation MethodsThese concepts are key in interpreting diagnostic tests and involve a mixture of basic probability and conditional probability formulas. For detailed calculations and practical applications, you can refer to the use of Bayes’ theorem and joint probability calculations, which are fundamental in medical statistics and epidemiology.### Useful Diagrams and Graphs#### Sensitivity and Specificity- **Sensitivity**: This is the probability that the test correctly identifies those with the condition (true positive rate).- **Specificity**: This is the probability that the test correctly identifies those without the condition (true negative rate).#### Confusion MatrixA confusion matrix can often be used to visualize these probabilities:- **True Positives (TP)**: Patients with hypertension correctly identified.- **True Negatives (TN)**: Patients without hypertension correctly identified.- **False Positives (FP)**: Patients without hypertension incorrectly identified as having

Given Proportion of certain pop has hypertension, p=0.22 Sensitivity=0.85 Specificity=0.91

2. A new diagnostic test is developed for detecting hypertension. Suppose that 22% of a certain population has hypertension. The sensitivity of the new test is 85% and the specificity is 91%. Suppose that a random subject from this population is selected. a. Find the probability that the subject has hypertension and tests positive. b. Find the probability that the subject does not have hypertension and tests negative. c. Find the probability that the subject has hypertension given the subject tests posi- tive. d. Find the probability that the subject does not have hypertension given the subject tests negative. e. Find the probability that the subject does not have hypertension given the subject tests positive.

2. A new diagnostic test is developed for detecting hypertension. Suppose that 22% of a certain population has hypertension. The sensitivity of the new test is 85% and the specificity is 91%. Suppose that a random subject from this population is selected. a. Find the probability that the subject has hypertension and tests positive. b. Find the probability that the subject does not have hypertension and tests negative. c. Find the probability that the subject has hypertension given the subject tests posi- tive. d. Find the probability that the subject does not have hypertension given the subject tests negative. e. Find the probability that the subject does not have hypertension given the subject tests positive.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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Question

I need help with a and b.

### Understanding Diagnostic Test Performance in Detecting Hypertension

A new diagnostic test is developed for detecting hypertension. This test is analyzed in a population where 22% of individuals have hypertension. The sensitivity (true positive rate) of the test is 85%, and its specificity (true negative rate) is 91%. Given these conditions, consider a random subject from this population. Here are some common probabilities one might want to calculate with the provided data:

#### Task Analysis

a. **Probability that the subject has hypertension and tests positive:**
- This is the joint probability of having hypertension and a positive test result. It can be calculated using the sensitivity of the test and the prevalence of hypertension in the population.

b. **Probability that the subject does not have hypertension and tests negative:**
- This involves calculating the joint probability of not having hypertension and receiving a negative test result, using the specificity of the test and the complement of the prevalence.

c. **Probability that the subject has hypertension given the subject tests positive:**
- This is a conditional probability problem. It involves Bayes’ theorem to determine the likelihood of having hypertension given a positive test result.

d. **Probability that the subject does not have hypertension given the subject tests negative:**
- This scenario also uses conditional probability to find the likelihood of not having hypertension given a negative test result.

e. **Probability that the subject does not have hypertension given the subject tests positive:**
- This examines the opposite condition of part (c), figuring out the likelihood of being healthy despite a positive test result.

### Calculation Methods

These concepts are key in interpreting diagnostic tests and involve a mixture of basic probability and conditional probability formulas. For detailed calculations and practical applications, you can refer to the use of Bayes’ theorem and joint probability calculations, which are fundamental in medical statistics and epidemiology.

### Useful Diagrams and Graphs

#### Sensitivity and Specificity
- **Sensitivity**: This is the probability that the test correctly identifies those with the condition (true positive rate).
- **Specificity**: This is the probability that the test correctly identifies those without the condition (true negative rate).

#### Confusion Matrix
A confusion matrix can often be used to visualize these probabilities:
- **True Positives (TP)**: Patients with hypertension correctly identified.
- **True Negatives (TN)**: Patients without hypertension correctly identified.
- **False Positives (FP)**: Patients without hypertension incorrectly identified as having

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