Questions:1. Suppose a person gets tested and receives a positive test result. We want to determine: with what probability can we conclude the person actually has the condition, given that they received a positive test result? The answer is not 95%. To investigate this question, it will be helpful to create a contingency table. Label the rows as + and - (the test result outcome) and label the columns "has it" and "does not have it" (the actual condition of the person in relation to the ailment). You can assume we have a random sample of, say, 1,000,000 individuals. From this you know what percentage are expected to have it. Then, you can use the accuracy of the test to figure out how many positive and negative results we expect. Once your table is complete, you can answer the above question. Why do you think the answer is so much lower than the 95%? 2. Ideally, everyone who gets a positive result is retested. We know from our table in Problem 1 how many people tested positive. Some of those people do have the condition and some don't. Nonetheless, they are frightened and are told by their doctor they will need to be retested. Recalculate the table with these new figures (there are no longer 1,000,000 people being tested). With what probability can we conclude the person actually has the condition, given that they received a positive test result again? Would you recommend for a third retest? DB Team Lead Required Question (these questions are optional if you are not the DB Lead this week, though I encourage you to engage in the challenge!):3. What we have observed as these hauntingly low probabilities is somewhat of an effect of the fact that the condition is somewhat "rare" - it only affects 0.9% of the population. Suppose we considered an ailment that affects 5% of the U.S. population. The test is again 95% accurate. Repeat questions 1 and 2 for this new scenario. What is your conclusion about what happens when we hold the accuracy of the test constant and we consider higher prevalence rates? (NOTE: prevalence is just a fancy way of describing what percentage of people get the condition.) 4. Congratulations! You are approaching a level of medical testing expertise most people do not have! One final thought question: since accuracy is such a confusing idea, why do you think we report accuracy instead of as the likelihood that a person actually has it, given that they get a positive test result? Use your observations in the problems above to help you think through this. Last week we learned that a test that is 99% accurate can be deceiving. Receivinga positive test result does not mean that we can be 99% confident in the result.In this week's discussion we will further investigate this idea by asking thequestion: how confident can we be if someone takes the test again and gets thesame result as on the first test?The definition of "accuracy" of a medical test is deceiving and confusing for mostpeople, doctors inclusive! As you learned in the Week 3 video example for amedical test, a positive test result is not often indicative of presence of acondition.When we learn that a test is 99% accurate, we are really supposed to derive fromthat these facts:If 100 people actually have the condition, then 99 of those people areexpected to get a positive test result. This is called a true positive. The oneperson who gets a negative test result even though they actually do have thecondition is said to have received a false negative.• If 100 people do not have the condition, then 99 of those people are expectedto get a negative test result. This is called a true negative. The one personwho received a positive test result even though they do not have thecondition is said to have received a false positive. Scientifically, this accuracy is determined by placing, say, bacteria on a dish andthen using the test to see if it detects the presence of the bacteria. A good testwill be able to detect the presence of that bacteria most of the time. Similarly, wewant to make sure the test isn't falsely detecting something that isn't there, so werepeat the experiment by testing dishes without the bacteria. We would expect agood test to indicate a negative most of the time.The issue with this is that we simply do not know whether or not a person has thecondition when getting tested. All we have to go off is the test result itself. This iswhere things get interesting. If the person gets a positive, how likely is it theyactually have it? If a person gets a negative, how likely is it that they don't? The99% does not directly apply to answering these questions, since the conditionalportion of our probability statement is "given a positive test result".Let's imagine we have some condition that ails 0.9% of the U.S. population, let'scall it Mathophobia. A test is developed that is 95% accurate. That is, the test willgive a positive result 95% of the time if we know that the bacteria is present.Similarly, it will give a negative result 95% of the time if we know the bacteria isnot present.Questions:1. Suppose a person gets tested and receives a positive test result. We wantto determine: with what probability can we conclude the person actuallyhas the condition, given that they received a positive test result? Theanswer is not 95%. To investigate this question, it will be helpful to create acontingency table. Label the rows as + and - (the test result outcome) andlabel the columns "has it" and "does not have it" (the actual condition of theperson in relation to the ailment). You can assume we have a randomsample of, say, 1,000,000 individuals. From this you know what percentageare expected to have it. Then, you can use the accuracy of the test to figureout how many positive and negative results we expect. Once your table iscomplete, you can answer the above question. Why do you think theanswer is so much lower than the 95%?

Answered: 1. Suppose a person gets tested and…

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Questions:

1. Suppose a person gets tested and receives a positive test result. We want to determine: with what probability can we conclude the person actually has the condition, given that they received a positive test result? The answer is not 95%. To investigate this question, it will be helpful to create a contingency table. Label the rows as + and - (the test result outcome) and label the columns "has it" and "does not have it" (the actual condition of the person in relation to the ailment). You can assume we have a random sample of, say, 1,000,000 individuals. From this you know what percentage are expected to have it. Then, you can use the accuracy of the test to figure out how many positive and negative results we expect. Once your table is complete, you can answer the above question. Why do you think the answer is so much lower than the 95%?

2. Ideally, everyone who gets a positive result is retested. We know from our table in Problem 1 how many people tested positive. Some of those people do have the condition and some don't. Nonetheless, they are frightened and are told by their doctor they will need to be retested. Recalculate the table with these new figures (there are no longer 1,000,000 people being tested). With what probability can we conclude the person actually has the condition, given that they received a positive test result again? Would you recommend for a third retest?

DB Team Lead Required Question (these questions are optional if you are not the DB Lead this week, though I encourage you to engage in the challenge!):

3. What we have observed as these hauntingly low probabilities is somewhat of an effect of the fact that the condition is somewhat "rare" - it only affects 0.9% of the population. Suppose we considered an ailment that affects 5% of the U.S. population. The test is again 95% accurate. Repeat questions 1 and 2 for this new scenario. What is your conclusion about what happens when we hold the accuracy of the test constant and we consider higher prevalence rates? (NOTE: prevalence is just a fancy way of describing what percentage of people get the condition.)

4. Congratulations! You are approaching a level of medical testing expertise most people do not have! One final thought question: since accuracy is such a confusing idea, why do you think we report accuracy instead of as the likelihood that a person actually has it, given that they get a positive test result? Use your observations in the problems above to help you think through this.

Last week we learned that a test that is 99% accurate can be deceiving. Receiving
a positive test result does not mean that we can be 99% confident in the result.
In this week's discussion we will further investigate this idea by asking the
question: how confident can we be if someone takes the test again and gets the
same result as on the first test?
The definition of "accuracy" of a medical test is deceiving and confusing for most
people, doctors inclusive! As you learned in the Week 3 video example for a
medical test, a positive test result is not often indicative of presence of a
condition.
When we learn that a test is 99% accurate, we are really supposed to derive from
that these facts:
If 100 people actually have the condition, then 99 of those people are
expected to get a positive test result. This is called a true positive. The one
person who gets a negative test result even though they actually do have the
condition is said to have received a false negative.
• If 100 people do not have the condition, then 99 of those people are expected
to get a negative test result. This is called a true negative. The one person
who received a positive test result even though they do not have the
condition is said to have received a false positive.

Scientifically, this accuracy is determined by placing, say, bacteria on a dish and
then using the test to see if it detects the presence of the bacteria. A good test
will be able to detect the presence of that bacteria most of the time. Similarly, we
want to make sure the test isn't falsely detecting something that isn't there, so we
repeat the experiment by testing dishes without the bacteria. We would expect a
good test to indicate a negative most of the time.
The issue with this is that we simply do not know whether or not a person has the
condition when getting tested. All we have to go off is the test result itself. This is
where things get interesting. If the person gets a positive, how likely is it they
actually have it? If a person gets a negative, how likely is it that they don't? The
99% does not directly apply to answering these questions, since the conditional
portion of our probability statement is "given a positive test result".
Let's imagine we have some condition that ails 0.9% of the U.S. population, let's
call it Mathophobia. A test is developed that is 95% accurate. That is, the test will
give a positive result 95% of the time if we know that the bacteria is present.
Similarly, it will give a negative result 95% of the time if we know the bacteria is
not present.
Questions:
1. Suppose a person gets tested and receives a positive test result. We want
to determine: with what probability can we conclude the person actually
has the condition, given that they received a positive test result? The
answer is not 95%. To investigate this question, it will be helpful to create a
contingency table. Label the rows as + and - (the test result outcome) and
label the columns "has it" and "does not have it" (the actual condition of the
person in relation to the ailment). You can assume we have a random
sample of, say, 1,000,000 individuals. From this you know what percentage
are expected to have it. Then, you can use the accuracy of the test to figure
out how many positive and negative results we expect. Once your table is
complete, you can answer the above question. Why do you think the
answer is so much lower than the 95%?

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