A diagnostic test for a certain disease is applied to n individuals known to not have the disease. Let X = the number among the n test results that are positive (indicating presence of the disease, so X is the number of false positives) and p = the probability that a disease-free individual's test result is positive (i.e., p is the true proportion of test results from disease-free individuals that are positive). Assume that only X is available rather than the actual sequence of test results. (a) Derive the maximum likelihood estimator of p. p= If n = 20 and x = 7, what is the estimate? p= (b) Is the estimator of part (a) unbiased? Yes No (c) If n = 20 and x = 7, what is the mle of the probability (1 - p)5 that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.)
A diagnostic test for a certain disease is applied to n individuals known to not have the disease. Let X = the number among the n test results that are positive (indicating presence of the disease, so X is the number of false positives) and p = the probability that a disease-free individual's test result is positive (i.e., p is the true proportion of test results from disease-free individuals that are positive). Assume that only X is available rather than the actual sequence of test results. (a) Derive the maximum likelihood estimator of p. p= If n = 20 and x = 7, what is the estimate? p= (b) Is the estimator of part (a) unbiased? Yes No (c) If n = 20 and x = 7, what is the mle of the probability (1 - p)5 that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.)
MATLAB: An Introduction with Applications
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![### Understanding Maximum Likelihood Estimation in Diagnostic Tests
A diagnostic test for a certain disease is applied to \( n \) individuals known to not have the disease. Let \( X \) be the number among the \( n \) test results that are positive (indicating the presence of the disease, so \( X \) is the number of false positives) and \( p \) be the probability that a disease-free individual’s test result is positive (i.e., \( p \) is the true proportion of test results from disease-free individuals that are positive). Assume that only \( X \) is available rather than the actual sequence of test results.
#### (a) Derive the Maximum Likelihood Estimator of \( p \).
\[ \hat{p} = \]
#### If \( n = 20 \) and \( x = 7 \), what is the estimate?
\[ \hat{p} = \]
#### (b) Is the estimator of part (a) unbiased?
- [ ] Yes
- [ ] No
#### (c) If \( n = 20 \) and \( x = 7 \), what is the MLE of the probability \( (1 - p)^5 \) that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.)
\[ (1 - \hat{p})^5 = \]
### Explanation of Concepts
**1. Maximum Likelihood Estimation (MLE):**
- The Maximum Likelihood Estimation (MLE) is a method used to estimate the parameters of a statistical model. The MLE is founded on finding the parameter values that maximize the likelihood that the process described by the model produced the data that were actually observed.
**2. False Positives:**
- In the context of the diagnostic test, a false positive occurs when the test incorrectly indicates the presence of the disease in an individual who does not have the disease.
**3. Unbiased Estimator:**
- An unbiased estimator is a statistic used to estimate a parameter that, on average, equals the true parameter value.
**4. Probability of No Positive Results in Future Tests:**
- This part involves calculating the likelihood that none of the next five tests will return a positive result, given the estimated probability of a single test being positive.
This exercise demonstrates how statistical methods can be applied to interpret and analyze the accuracy and reliability of diagnostic tests.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb989860-4d1c-4fb5-9e6b-42a4528dce9c%2Fdc3779b8-7f07-48f9-8dc1-81b36e9a99ca%2Fp8phb7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Maximum Likelihood Estimation in Diagnostic Tests
A diagnostic test for a certain disease is applied to \( n \) individuals known to not have the disease. Let \( X \) be the number among the \( n \) test results that are positive (indicating the presence of the disease, so \( X \) is the number of false positives) and \( p \) be the probability that a disease-free individual’s test result is positive (i.e., \( p \) is the true proportion of test results from disease-free individuals that are positive). Assume that only \( X \) is available rather than the actual sequence of test results.
#### (a) Derive the Maximum Likelihood Estimator of \( p \).
\[ \hat{p} = \]
#### If \( n = 20 \) and \( x = 7 \), what is the estimate?
\[ \hat{p} = \]
#### (b) Is the estimator of part (a) unbiased?
- [ ] Yes
- [ ] No
#### (c) If \( n = 20 \) and \( x = 7 \), what is the MLE of the probability \( (1 - p)^5 \) that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.)
\[ (1 - \hat{p})^5 = \]
### Explanation of Concepts
**1. Maximum Likelihood Estimation (MLE):**
- The Maximum Likelihood Estimation (MLE) is a method used to estimate the parameters of a statistical model. The MLE is founded on finding the parameter values that maximize the likelihood that the process described by the model produced the data that were actually observed.
**2. False Positives:**
- In the context of the diagnostic test, a false positive occurs when the test incorrectly indicates the presence of the disease in an individual who does not have the disease.
**3. Unbiased Estimator:**
- An unbiased estimator is a statistic used to estimate a parameter that, on average, equals the true parameter value.
**4. Probability of No Positive Results in Future Tests:**
- This part involves calculating the likelihood that none of the next five tests will return a positive result, given the estimated probability of a single test being positive.
This exercise demonstrates how statistical methods can be applied to interpret and analyze the accuracy and reliability of diagnostic tests.
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