Question

In: Statistics and Probability

In a certain community, 4 percent of all adults over the age of 50 have tuberculosis...

In a certain community, 4 percent of all adults over the age of 50 have tuberculosis (T.B.). A health service in this community correctly diagnoses 98 percent of all persons with T.B. as having the disease, and incorrectly diagnoses 3 percent of all persons without T.B. as having the disease. Find the probabilities that,

a) the community health service will diagnose an adult over 50 as having T.B.,

b) a person over 50 diagnosed by the health service as having T.B. actually has the disease.

Solutions

Expert Solution

Let, A be the event that the person is diagnosed with TB

Let, E1 be the event that person actually has TB

Let, E2 be the event that person does not have TB

Probability person having TB , P(E1) = 4% = 0.04

Probability person does not have TB , P(E2) = 1 - P(E1) = 1 - 0.04 = 0.96

Probability person having TB and diagnosed correctly, P[ A | E1 ] = 98% = 0.98

Probability person does not have TB and diagnosed incorrectly, P[ A | E1 ] = 98% = 3% = 0.03

a) Probability that the community health service will diagnose an adult over 50 as having T.B, P(A)

P(A) = P(E1)*P[ A | E1 ] +  P(E2)*P[ A | E2 ] = 0.04*0.98 + 0.96*0.03 = 0.0392 + 0.0288 = 0.068 = 6.8%

P(A) = 0.068 = 6.8%

b) Probability a person over 50 diagnosed by the health service as having T.B. actually has the disease = P[ A | E1 ]

P[ A | E1 ] = P(E1)*P[ A | E1 ] / P(A) ( using Baye's theorem )

P[ A | E1 ] = 0.04*0.98 / 0.068 = 0.0392 / 0.068 = 0.5765

P[ A | E1 ] = 0.5765 = 57.65%


Related Solutions

In a certain community, 4 percent of all adults over the age of 50 have tuberculosis...
In a certain community, 4 percent of all adults over the age of 50 have tuberculosis (T.B.). A health service in this community correctly diagnoses 98 percent of all persons with T.B. as having the disease, and incorrectly diagnoses 3 percent of all persons without T.B. as having the disease. Find the probabilities that, a) the community health service will diagnose an adult over 50 as having T.B., b) a person over 50 diagnosed by the health service as having...
A recent survey found that 70% of all adults over 50 wear glasses for driving. In...
A recent survey found that 70% of all adults over 50 wear glasses for driving. In a random sample of 10 adults over 50, what is the probability that exactly six wear glasses? Please round answer to the ten-thousandths place (4 places).
In a certain region, 20% of people over age 50 didn't graduate from high school. We...
In a certain region, 20% of people over age 50 didn't graduate from high school. We would like to know if this percentage is the same among the 25-30 year age group. Use critical values to exactly 3 decimal places. (a) How many 25-30 year old people should be surveyed in order to estimate the proportion of non-grads to within 7% with 95% confidence? (b) Suppose we wanted to cut the margin of error to 4%. How many people should...
In a certain region, 24% of people over age 50 didn't graduate from high school. We...
In a certain region, 24% of people over age 50 didn't graduate from high school. We would like to know if this percentage is the same among the 25-30 year age group. Use critical values to exactly 3 decimal places. (a) How many 25-30 year old people should be surveyed in order to estimate the proportion of non-grads to within 6% with 99% confidence? (b) Suppose we wanted to cut the margin of error to 2%. How many people should...
A recent survey found that 63% of all adults over 50 wear glasses for driving. Ia...
A recent survey found that 63% of all adults over 50 wear glasses for driving. Ia random sample of 100 adults over 50 was selected., a). what is the expected value and standard deviation of the number of adults over 50 who wear glasses? b). Find the probability that more than 60 of the adults over 50 wear glasses. Hint: using normal distribution to approximate binomial distribution.
It is believed that 1/4 of adults over 50 years old graduated from high school. We...
It is believed that 1/4 of adults over 50 years old graduated from high school. We want to see if the percentage is the same for 30 year olds. How many should we survey in order to estimate the proportion of non-grads within 0.06 with 90 % confidence? Suppose we have no idea of the % of non-grads, what would be the "n" value?
A stock had returns of 10 percent, -4 percent, 4 percent, and 18 percent over the...
A stock had returns of 10 percent, -4 percent, 4 percent, and 18 percent over the past four years. What is the standard deviation of these returns?
8. Tuberculosis The genus mycobacterium contains over 50 species with several human pathogens of concern. Mycobacteria...
8. Tuberculosis The genus mycobacterium contains over 50 species with several human pathogens of concern. Mycobacteria are distinguishable from other types of bacteria by the presence of wax layers and high molecular weight fatty acids (mycolic acids). This complex, external structure offers protection from acids, drying and some germicides. In fact, mycobacteria are also referred to as acid-fast bacilli because acid treatments will not result in decolorization during staining. One of the mycobacteria species of medical interest is Mycobacterium tuberculosis,...
A community health nurse is cut is discussing the self concept characteristics with middle age adults...
A community health nurse is cut is discussing the self concept characteristics with middle age adults that occur during this stage of life. Discuss 3 characteristics for self concept of concern for this stage of life, depression, irritability, and menopause.
a community health nurse is discussing body image with middle age adults. discuss the characteristics for...
a community health nurse is discussing body image with middle age adults. discuss the characteristics for body image sex drive, churned in physical appearance, and women: symptoms of menopause can represent as new interest in intimacy, or loss of the reproductive role or femininity
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT