need help on stats 243 project

Project Conditional (non independent) Probability COMID's been pushing us around for (as write this just about 2 years. And I know that we're all sick as hell of hearing about but it's a great application of a really, really neat idea in probability. Let's paint yaa piure to motivate it Suppose you got to get tested for COVID, and the test comes back positive 1. (2 points) What is the chance, do you think that you have it? the a scale of 0 to 100%, "" meaning "There's no way have it to "100%" meaning "absolutely have you're just estimating here, based on your intuition, we'll be working out the numbers soon! No anowers are incorrect Also write a sentence or two about why you answered the way you did Now, in perfect world, if you tested positive for something that would mean you had it, without a doubt. Conversely, testing negative for something would mean you wouldn't have. But we don't live in a perfect world we live in a world where tests are sometimes wrong-and tests ike these can be wrong in two different ways. But, they can also be right in two different ways Confused? Totally understandable. We walk through the posibilities here in a chart De you have COVAD Yes To CIS CS scaled as You' Positive that e COVE but the set tut Youtu COVID and comes back.. You COVE the AW COVD, and Negative data you th So, again in an ideal world, you'd want thour green boxes to happen all the time (depending on whether or not you have COVID) and you'd want the red boxes to happen wel, never but that's the problem: those false positive and negatives de tuppent "OK, Sean-what's the big deal? You might ask "So what you get a faise positive test result? The best comes back positive, it was wrong, and maybe you were a little inconvenienced because you had to-quarantine. Big deal!" 2 (2) What about a false negative COVIO test for would argue, a fale negative test for any infectious disease. Why can that be a huge deal? So, yeah, its a big deal to get the test right-but "getting it right means, actually, 4 different things (as we just saw Maxoming the chances of true positives, Maximizing the chances of tru Minimizing the chances of fase positives, and Minimizing the chances of fale negatives nghofTH 25 may not of the So, that's what we study in this question the mathematics of how some of the currently available COVID work! Now, part of what makes this question interesting and not at all trvabis that no one really know for sure, what the prevalence rate for COVID & in the hour or wo of my searching of various CDC sites (among different locales both national and international, I found that it appears as though the rate might average around 1%. So, what figure! So now, let's break the US down into say, mal groups of 100,000 people who are unfermly, randomly distributed). Assuming that indeed, the infection rate is 1N, that means that 1000 of them will get COMID, and the other 93000 won't Let's redraw that table from above with this new information De you have COVAD Yes 1000 OK! That's coming along nicely. No 99000 Maximing the chances of true positive, Maximizing the chances of true negatives, Minimizing the chances of false positives, and Minimizing the chances of false negatives in the of MTH 205 who muy gosh e treatment of the ab So, that's what we'll study in this question: the mathematics of how some of the currently available COVID tests Now, part of what makes this question interesting (and not at all trivia is that no one really know for sure, what the prevalence rate for COVID is in the hour or so of my searching of various CDC sites (among different locales- both national and internationall, I found that it appears as though the rate might average and 1%. So, we'l use the figure! So now, let's break the US down into, say, small groups of 100,000 people who are uniformly, randomly distributed) Ausuming that, indeed, the infection rate is 1%, that means that 1000 of them will get COVD, and the other 99000 won't Let's redraw that table from aboww with this new information! De you have COVAD Yes No 1000 99000 O That's coming along nicely But remember-all 100,000 of those people take a COVID test, some will come back positive, and the rest wil come back negative. And, in an ideal testing world, all the folks with COVID would get positive test results, and all the folks without it would get negative test results Do you have COMID Yes No Positive 1000 0 You take perfect COVID test in thi ideal w and it comes back... Negative 99000 country's COV can't enough to the 25% as an average infectionate as the up in "hot", and other places have vay lower numbers as type this actually On doing well Well with that mean, other than the fact that we're talking about a deadly panders. The math though But, the tests aren't perfect, and we don't live in an ideal world so we need to deal with false positives and false negatives I took a trip back out to the interwets and found that the faise positive rate for the most effective COVID test (the painful-sounding-but-not-too-awful "deep nose saab one was between 0.8% and 4% ers assume, for purposes of illustration, it's 2% (inds in the middle of that rangel So what that means is this: 2% of people who don't have COMID will test positive. And that means the chart would look more like this: Do you have COVID? Yes No 1980 COM 3 of 7 Of a talking about a deady sederic The huko god, But, the tests anew't perfect, and we don't live in an ideal world so we need to deal with tabe positives and tale negative took a trip back out to the interwebs and found that the false positive rate for the most effective COVID test (the painful-sounding-but-not-too-awfuldeep nose s one was between 0.8% and 4%-se's atsume, for purposes of illustration, 2% (nda in the middle of that range So what that means is this: 2% of people who don't here COMID will still best positive. And that means the chart would look more like this De you have COVO Yes No Positive 1980 (2% of that 99000) You take COVIO test and with the in member 100 it comes back... Negative 97020 (the other 98%) So, that's a wild-even with just a 2% fabe positive rate, you have amat 2000 people testing pouvel but that kind makes sense, too: most people don't have COVID. thankfully Ox-now it's time to deal with false negatives Remember, these are the ones we should really be neu about! According to more sites I was able to track down, the false negative rate is ay between 2% and 37% That 37% threw me for a loop-wicked hight But, it's an outier compared to the other one, so let's fa positive rate that's more toward the 2% say 5%. Then, we can extend our last chart to look th De you have COV Yes No Positive 950 1980 You take a COVID text and it comes back. Negative 50 97020 www..com/22 befor DANIND OK! Now we can look again at your answer from up there where you estimated your chance of having COVIO, assuming you tested postwel. We're going to figure out that chance right now, based on the above chart! Chance you have COVID, assuming you tested positive those who have COVID and tested positive all those who tested positive -32% 950-1980 Chance you don't have COVID, assuming you tested positive 1-Chance you don't have COVID, you tested positive - 1-0.32= 68% If tested positive for something like to be sure at least, mostly sh Alas, this is the reality of disease testing even though the percentages of fate positives and false negatives sound "prod" 12% and 56, respectively, they're not applied equally since, thankfully, most people don't have COVIDI fale pot OK! Now we can look again at your answer from #1 up there (where you estimated your chance of having COVID, assuming you tested positive). We're going to figure out that chance right now, based on the above chart! Chance you have COVID, assuming you tested positive those who have COVID and tested positive all those who tested positive 950 950-1980 32% Chance you don't have COVID, assuming you 1-Chance you don't have COVID, assuming 1-0.32 - 68% you tested positive tested positive If I tested positive for something, I'd like to be sure (at least, mostly sure) that I had it! Alas, this is the reality of disease testing even though the percentages of false positives and false negatives sound "good" (2% and 5%, respectively), they're not applied equally (since, thankfully, most people don't have COVID) But, remember..false positives, while annoying, aren't necessarily the ones I'm worried about. No..I'm more concerned about the false negatives (especially after reading what y'all put up here in #2). 3. (2 points)(w) Using the numbers in the above chart (and a similar approach as I did right below that chart), find the chance that, if someone tests negative for COVID, they don't have it. 4. (1 point) That last value was the chance of a true negative from a COVID test. What's the chance of a false negative? OK-84 is a small probability, which is good! But, still.. those people can then go out and wander around, potentially infecting those around them But, also important that that is the fact that we used "average case scenario" false positive and false negative rates. Remember from above that false positive rates can be as high as 4% and false negative rates can be as high as 37% Use these new "worst case scenario" values (let's use an infection rate of 2% for now) and find: 5. (3 points)(w) The chance that someone who tests positive actually has COMID you like, you can use this bansy dandy little tool built for you it'll get you the table like above; all you have to do is show me the fractions you create and give me the final percentagel 6 (3 points) (w) The chance that someone who tests negative actually doesn't have COVID Whoa! Now, a positive test means you're actually about 3 times more likely to not have COVID (while a negative test still means you're pretty likely to not have it, either). Woof-with those false positive and false negative values, it's really not even worth taking a test, honestly But remember! We assumed a disease prevalence rate of around 2%, but no one really knows how prevalent it is (nor how absolutely reliable the tests are). Of course, typing this about a month before you actually turs this in. But have the @ This kind of mathematics is one of the most wondrous examples of truly useful conditional probabilities (we looked at a few many weeks ago in your first project). Conditional probabilities are the cousins of independent ones with independent probabilities, one event's results was not influenced in any way by another; in conditional probabilities, the events are linked. By what, we might not know-but they're definitely not independent. Let's turn away from COVID now, and look at some examples where rates are a bit more solidified if you've ever spent time at lovely East Lake in Oregon, you might know about the mercury levels of some of the fish there". Environmentalists have developed a test for determining when the mercury level in fish is above "permissible levels" set by the FDA and EPA if the sampled fish's mercury level exceeds this permissible level, the test is 99% effective in determining this (which means that, 1% of the time, it will miss the mercury a "false negative"), However, if the fish's mercury level is within permissible limits, the best will say the fish is "OK" 96% of the time (meaning that, 4% of the time, the fish will get a "false positive"] Now, we also have to have a "prevalence of mercury number; I searched high and low for this, but came up empty, so I'll settle on 50% for starters (and scientists can refine that as they go). This kind of mathematics is one of the most wondrous examples of truly unetul conditional probabilities we looked at a few many weeks ago in your first project) Conditional probabilities are the cousins of independent ones with independent probabilities, one event's results was not influenced in any way by another, in conditional probabilities, the events are linked. By what we might not know-but they're definitely not independent Let's taway from COVID now, and look at some examples where rates are a bit more solidified If you've ever spent time at lovely East Lake in Oregon, you mighe know about the mercury levels of some of the fish there". Environmentalists have developed a test for determining when the mercury level in fish is above "permissible levels" set by the FDA and EPA, if the sampled fish's mercury level exceeds this permissible level, the test is 99% effective in determining this which means that, 1% of the time it will miss the mercury.a "Tale negative"). However, if the fish's mercury levels within permissible limits, the test will say the fish is "O 96% of the time (meaning that, 4% of the time, the fish will get a "faise positive) Now, we also have to have a "prevalence of mercury number, I searched high and low for this, but came up empty, soll set on 50% for starters (and scientists can refine that as they go If a fish tests positive for mercury, what is the probability that the fish's mercury level exceeds permisible levels? And what if the fish tests negative what's the chance's clean Ah, what the hello a little coding in Excel, shall we? Rad! So now, you have a template for running the numbers on any test! Lets give it a shet! (no need to show any spreadsheet stuff/work for these tasume you're doing them the way you did the ones above There is a test for Down's syndrome that can be done during the 4 month of pregnancy for "tik" mothers an at-risk fetus actually has (or will develop) Down's syndrome, the test is accurate 65% of the time meaning it has a 35% fale negative rate), but also carries a 7.2% fase positive rate. A pregnant friend of mine took the test, and the test came back positive (per the CDC, the incidence of Downs in the general population is 1 in 500 for her age group; it increases super-exponentially as the mother gets older) 7. (2 points) What is the chance that her baby will have Down's syndrome? &(2 points) How about the other way what if she tested negative? What would have been the chance her baby wouldn't have had Downs? When I shared these results with her, she felt more at ease. But, then she also asked me a great question.one that I had asked myself many times "Why the hell would you even do this test , no matter what the test says, there's a 99.9% chance of no Downy?" This is called "s a "Bayer Theo Welp, this is why these kinds of mathematical analyses are great there are three different things going on in each! The sensitivity of the test (that's a fancy word that medical folks sometimes use to mean "true positive The specificity of the test byen, you guessed-the "true negative rate) 50, looking above, that 65% accurate test at Ding Downs will be the sensitivity (the "35%" sabe negative because the fetus is at risk of developing Downs (that's the condition: .65% of the time, the test will say sobre positive") The other 33% of the time, it won't so it's abely negative" That test's specificity would be 52.8%, for the same reason of the fetus is not at risk of developing Downs 922% of the time, it will say "no Down The other 7.2%, it was falsely positively result Now, thoor numbers Sugged me, so I decided to look at some updated values found this study, which used a test with updated sensitivity and specificity values sensitivity: 97.25%, speccy 95. Let's use those! 91 point) What's the false positive rate for this new test? 10. (1point) What's the fale negative rate for this new? 11. (2 points) Repeat #7 using this new test (same incidence of Downs as up there Welp, this is why these kinds of mathematical analyses are great: there are three different things going on in each! The rate of whatever you're testing for. The sensitivity of the test (that's a fancy word that medical folks sometimes use to mean "true positive rate") The specificity of the test (yep, you guessed it-the "true negative rate") So, looking above, that "65% accurate test at ID-ing Downs" will be the sensitivity (the "35%" is a false negative because, if the fetus is at risk of developing Downs (that's the condition): . 65% of the time, the test will say so ("true positive"); The other 35% of the time, it won't (so it's "falsely negative") That test's specificity would be 92.8%, for the same reason: if the fetus is not at risk of developing Downs: . 92.8% of the time, it will say "no Downs" ("true negative"); The other 7.2%, it was falsely positively result. Now, those numbers bugged me, so I decided to look at some updated values. I found this study, which used a test with updated sensitivity and specificity values (sensitivity: 97.2%, specificity: 99.8%). Let's use those! 9. (1 point) What's the false positive rate for this new test? 10. (1 point) What's the false negative rate for this new test? 11. (2 points) Repeat #7 using this new test (same incidence of Downs as up there). See the difference a test can make? Now you're beginning to see why so many folks argue about COVID testing: it really depends on which test! If these kinds of analyses are making your head spin, you're not alone. Even doctors screw this stuff up Gerd Gigerenzer, a cognitive psychologist at the Max Planck Institute for Human Development in Berlin, asked doctors in Germany and the United States to estimate the probability that a woman in the general population with a positive mammogram actually has breast cancer. Here's the language he used: "A 50-year-old woman, no symptoms, participates in routine mammography screening. She tests positive, is alarmed, and wants to know from you whether she has breast cancer for certain or what the chances are. Apart from the screening results, you know nothing else about this woman. What percent of women who test positive actually have breast cancer? What is the best answer? nine in 10 eight in 10. one in 10 one in 100 12. (1 point) Shoot from the hip first: which one of those would you pick off the bat, without doing any math? GG then supplied the doctors with this, to assist them: This is why so glad there are so many nursing students in my stat classes. Someone has to keep these damn doctors in line with a totally rad name "The probability that a randomly selected woman has breast cancer is about 0.8% (that is, 0.008). If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 9 percent that she will have a falsely positive mammogram. 13. (3 points) OK- now do the math cancer actually has it? What's the probability that a woman who testes positive for breast OK, GG.. how'd the docs do? "In one session, almost half the group of 160 gynecologists responded that the woman's chance of having cancer was nine in 10. Only 21% said that the figure was one in 10-which is the correct answer. That's a worse result than if the doctors had been answering at random. See? This is why I'm excited you're here in MTH 243. . . Project Conditional (non independent) Probability COMID's been pushing us around for (as write this just about 2 years. And I know that we're all sick as hell of hearing about but it's a great application of a really, really neat idea in probability. Let's paint yaa piure to motivate it Suppose you got to get tested for COVID, and the test comes back positive 1. (2 points) What is the chance, do you think that you have it? the a scale of 0 to 100%, "" meaning "There's no way have it to "100%" meaning "absolutely have you're just estimating here, based on your intuition, we'll be working out the numbers soon! No anowers are incorrect Also write a sentence or two about why you answered the way you did Now, in perfect world, if you tested positive for something that would mean you had it, without a doubt. Conversely, testing negative for something would mean you wouldn't have. But we don't live in a perfect world we live in a world where tests are sometimes wrong-and tests ike these can be wrong in two different ways. But, they can also be right in two different ways Confused? Totally understandable. We walk through the posibilities here in a chart De you have COVAD Yes To CIS CS scaled as You' Positive that e COVE but the set tut Youtu COVID and comes back.. You COVE the AW COVD, and Negative data you th So, again in an ideal world, you'd want thour green boxes to happen all the time (depending on whether or not you have COVID) and you'd want the red boxes to happen wel, never but that's the problem: those false positive and negatives de tuppent "OK, Sean-what's the big deal? You might ask "So what you get a faise positive test result? The best comes back positive, it was wrong, and maybe you were a little inconvenienced because you had to-quarantine. Big deal!" 2 (2) What about a false negative COVIO test for would argue, a fale negative test for any infectious disease. Why can that be a huge deal? So, yeah, its a big deal to get the test right-but "getting it right means, actually, 4 different things (as we just saw Maxoming the chances of true positives, Maximizing the chances of tru Minimizing the chances of fase positives, and Minimizing the chances of fale negatives nghofTH 25 may not of the So, that's what we study in this question the mathematics of how some of the currently available COVID work! Now, part of what makes this question interesting and not at all trvabis that no one really know for sure, what the prevalence rate for COVID & in the hour or wo of my searching of various CDC sites (among different locales both national and international, I found that it appears as though the rate might average around 1%. So, what figure! So now, let's break the US down into say, mal groups of 100,000 people who are unfermly, randomly distributed). Assuming that indeed, the infection rate is 1N, that means that 1000 of them will get COMID, and the other 93000 won't Let's redraw that table from above with this new information De you have COVAD Yes 1000 OK! That's coming along nicely. No 99000 Maximing the chances of true positive, Maximizing the chances of true negatives, Minimizing the chances of false positives, and Minimizing the chances of false negatives in the of MTH 205 who muy gosh e treatment of the ab So, that's what we'll study in this question: the mathematics of how some of the currently available COVID tests Now, part of what makes this question interesting (and not at all trivia is that no one really know for sure, what the prevalence rate for COVID is in the hour or so of my searching of various CDC sites (among different locales- both national and internationall, I found that it appears as though the rate might average and 1%. So, we'l use the figure! So now, let's break the US down into, say, small groups of 100,000 people who are uniformly, randomly distributed) Ausuming that, indeed, the infection rate is 1%, that means that 1000 of them will get COVD, and the other 99000 won't Let's redraw that table from aboww with this new information! De you have COVAD Yes No 1000 99000 O That's coming along nicely But remember-all 100,000 of those people take a COVID test, some will come back positive, and the rest wil come back negative. And, in an ideal testing world, all the folks with COVID would get positive test results, and all the folks without it would get negative test results Do you have COMID Yes No Positive 1000 0 You take perfect COVID test in thi ideal w and it comes back... Negative 99000 country's COV can't enough to the 25% as an average infectionate as the up in "hot", and other places have vay lower numbers as type this actually On doing well Well with that mean, other than the fact that we're talking about a deadly panders. The math though But, the tests aren't perfect, and we don't live in an ideal world so we need to deal with false positives and false negatives I took a trip back out to the interwets and found that the faise positive rate for the most effective COVID test (the painful-sounding-but-not-too-awful "deep nose saab one was between 0.8% and 4% ers assume, for purposes of illustration, it's 2% (inds in the middle of that rangel So what that means is this: 2% of people who don't have COMID will test positive. And that means the chart would look more like this: Do you have COVID? Yes No 1980 COM 3 of 7 Of a talking about a deady sederic The huko god, But, the tests anew't perfect, and we don't live in an ideal world so we need to deal with tabe positives and tale negative took a trip back out to the interwebs and found that the false positive rate for the most effective COVID test (the painful-sounding-but-not-too-awfuldeep nose s one was between 0.8% and 4%-se's atsume, for purposes of illustration, 2% (nda in the middle of that range So what that means is this: 2% of people who don't here COMID will still best positive. And that means the chart would look more like this De you have COVO Yes No Positive 1980 (2% of that 99000) You take COVIO test and with the in member 100 it comes back... Negative 97020 (the other 98%) So, that's a wild-even with just a 2% fabe positive rate, you have amat 2000 people testing pouvel but that kind makes sense, too: most people don't have COVID. thankfully Ox-now it's time to deal with false negatives Remember, these are the ones we should really be neu about! According to more sites I was able to track down, the false negative rate is ay between 2% and 37% That 37% threw me for a loop-wicked hight But, it's an outier compared to the other one, so let's fa positive rate that's more toward the 2% say 5%. Then, we can extend our last chart to look th De you have COV Yes No Positive 950 1980 You take a COVID text and it comes back. Negative 50 97020 www..com/22 befor DANIND OK! Now we can look again at your answer from up there where you estimated your chance of having COVIO, assuming you tested postwel. We're going to figure out that chance right now, based on the above chart! Chance you have COVID, assuming you tested positive those who have COVID and tested positive all those who tested positive -32% 950-1980 Chance you don't have COVID, assuming you tested positive 1-Chance you don't have COVID, you tested positive - 1-0.32= 68% If tested positive for something like to be sure at least, mostly sh Alas, this is the reality of disease testing even though the percentages of fate positives and false negatives sound "prod" 12% and 56, respectively, they're not applied equally since, thankfully, most people don't have COVIDI fale pot OK! Now we can look again at your answer from #1 up there (where you estimated your chance of having COVID, assuming you tested positive). We're going to figure out that chance right now, based on the above chart! Chance you have COVID, assuming you tested positive those who have COVID and tested positive all those who tested positive 950 950-1980 32% Chance you don't have COVID, assuming you 1-Chance you don't have COVID, assuming 1-0.32 - 68% you tested positive tested positive If I tested positive for something, I'd like to be sure (at least, mostly sure) that I had it! Alas, this is the reality of disease testing even though the percentages of false positives and false negatives sound "good" (2% and 5%, respectively), they're not applied equally (since, thankfully, most people don't have COVID) But, remember..false positives, while annoying, aren't necessarily the ones I'm worried about. No..I'm more concerned about the false negatives (especially after reading what y'all put up here in #2). 3. (2 points)(w) Using the numbers in the above chart (and a similar approach as I did right below that chart), find the chance that, if someone tests negative for COVID, they don't have it. 4. (1 point) That last value was the chance of a true negative from a COVID test. What's the chance of a false negative? OK-84 is a small probability, which is good! But, still.. those people can then go out and wander around, potentially infecting those around them But, also important that that is the fact that we used "average case scenario" false positive and false negative rates. Remember from above that false positive rates can be as high as 4% and false negative rates can be as high as 37% Use these new "worst case scenario" values (let's use an infection rate of 2% for now) and find: 5. (3 points)(w) The chance that someone who tests positive actually has COMID you like, you can use this bansy dandy little tool built for you it'll get you the table like above; all you have to do is show me the fractions you create and give me the final percentagel 6 (3 points) (w) The chance that someone who tests negative actually doesn't have COVID Whoa! Now, a positive test means you're actually about 3 times more likely to not have COVID (while a negative test still means you're pretty likely to not have it, either). Woof-with those false positive and false negative values, it's really not even worth taking a test, honestly But remember! We assumed a disease prevalence rate of around 2%, but no one really knows how prevalent it is (nor how absolutely reliable the tests are). Of course, typing this about a month before you actually turs this in. But have the @ This kind of mathematics is one of the most wondrous examples of truly useful conditional probabilities (we looked at a few many weeks ago in your first project). Conditional probabilities are the cousins of independent ones with independent probabilities, one event's results was not influenced in any way by another; in conditional probabilities, the events are linked. By what, we might not know-but they're definitely not independent. Let's turn away from COVID now, and look at some examples where rates are a bit more solidified if you've ever spent time at lovely East Lake in Oregon, you might know about the mercury levels of some of the fish there". Environmentalists have developed a test for determining when the mercury level in fish is above "permissible levels" set by the FDA and EPA if the sampled fish's mercury level exceeds this permissible level, the test is 99% effective in determining this (which means that, 1% of the time, it will miss the mercury a "false negative"), However, if the fish's mercury level is within permissible limits, the best will say the fish is "OK" 96% of the time (meaning that, 4% of the time, the fish will get a "false positive"] Now, we also have to have a "prevalence of mercury number; I searched high and low for this, but came up empty, so I'll settle on 50% for starters (and scientists can refine that as they go). This kind of mathematics is one of the most wondrous examples of truly unetul conditional probabilities we looked at a few many weeks ago in your first project) Conditional probabilities are the cousins of independent ones with independent probabilities, one event's results was not influenced in any way by another, in conditional probabilities, the events are linked. By what we might not know-but they're definitely not independent Let's taway from COVID now, and look at some examples where rates are a bit more solidified If you've ever spent time at lovely East Lake in Oregon, you mighe know about the mercury levels of some of the fish there". Environmentalists have developed a test for determining when the mercury level in fish is above "permissible levels" set by the FDA and EPA, if the sampled fish's mercury level exceeds this permissible level, the test is 99% effective in determining this which means that, 1% of the time it will miss the mercury.a "Tale negative"). However, if the fish's mercury levels within permissible limits, the test will say the fish is "O 96% of the time (meaning that, 4% of the time, the fish will get a "faise positive) Now, we also have to have a "prevalence of mercury number, I searched high and low for this, but came up empty, soll set on 50% for starters (and scientists can refine that as they go If a fish tests positive for mercury, what is the probability that the fish's mercury level exceeds permisible levels? And what if the fish tests negative what's the chance's clean Ah, what the hello a little coding in Excel, shall we? Rad! So now, you have a template for running the numbers on any test! Lets give it a shet! (no need to show any spreadsheet stuff/work for these tasume you're doing them the way you did the ones above There is a test for Down's syndrome that can be done during the 4 month of pregnancy for "tik" mothers an at-risk fetus actually has (or will develop) Down's syndrome, the test is accurate 65% of the time meaning it has a 35% fale negative rate), but also carries a 7.2% fase positive rate. A pregnant friend of mine took the test, and the test came back positive (per the CDC, the incidence of Downs in the general population is 1 in 500 for her age group; it increases super-exponentially as the mother gets older) 7. (2 points) What is the chance that her baby will have Down's syndrome? &(2 points) How about the other way what if she tested negative? What would have been the chance her baby wouldn't have had Downs? When I shared these results with her, she felt more at ease. But, then she also asked me a great question.one that I had asked myself many times "Why the hell would you even do this test , no matter what the test says, there's a 99.9% chance of no Downy?" This is called "s a "Bayer Theo Welp, this is why these kinds of mathematical analyses are great there are three different things going on in each! The sensitivity of the test (that's a fancy word that medical folks sometimes use to mean "true positive The specificity of the test byen, you guessed-the "true negative rate) 50, looking above, that 65% accurate test at Ding Downs will be the sensitivity (the "35%" sabe negative because the fetus is at risk of developing Downs (that's the condition: .65% of the time, the test will say sobre positive") The other 33% of the time, it won't so it's abely negative" That test's specificity would be 52.8%, for the same reason of the fetus is not at risk of developing Downs 922% of the time, it will say "no Down The other 7.2%, it was falsely positively result Now, thoor numbers Sugged me, so I decided to look at some updated values found this study, which used a test with updated sensitivity and specificity values sensitivity: 97.25%, speccy 95. Let's use those! 91 point) What's the false positive rate for this new test? 10. (1point) What's the fale negative rate for this new? 11. (2 points) Repeat #7 using this new test (same incidence of Downs as up there Welp, this is why these kinds of mathematical analyses are great: there are three different things going on in each! The rate of whatever you're testing for. The sensitivity of the test (that's a fancy word that medical folks sometimes use to mean "true positive rate") The specificity of the test (yep, you guessed it-the "true negative rate") So, looking above, that "65% accurate test at ID-ing Downs" will be the sensitivity (the "35%" is a false negative because, if the fetus is at risk of developing Downs (that's the condition): . 65% of the time, the test will say so ("true positive"); The other 35% of the time, it won't (so it's "falsely negative") That test's specificity would be 92.8%, for the same reason: if the fetus is not at risk of developing Downs: . 92.8% of the time, it will say "no Downs" ("true negative"); The other 7.2%, it was falsely positively result. Now, those numbers bugged me, so I decided to look at some updated values. I found this study, which used a test with updated sensitivity and specificity values (sensitivity: 97.2%, specificity: 99.8%). Let's use those! 9. (1 point) What's the false positive rate for this new test? 10. (1 point) What's the false negative rate for this new test? 11. (2 points) Repeat #7 using this new test (same incidence of Downs as up there). See the difference a test can make? Now you're beginning to see why so many folks argue about COVID testing: it really depends on which test! If these kinds of analyses are making your head spin, you're not alone. Even doctors screw this stuff up Gerd Gigerenzer, a cognitive psychologist at the Max Planck Institute for Human Development in Berlin, asked doctors in Germany and the United States to estimate the probability that a woman in the general population with a positive mammogram actually has breast cancer. Here's the language he used: "A 50-year-old woman, no symptoms, participates in routine mammography screening. She tests positive, is alarmed, and wants to know from you whether she has breast cancer for certain or what the chances are. Apart from the screening results, you know nothing else about this woman. What percent of women who test positive actually have breast cancer? What is the best answer? nine in 10 eight in 10. one in 10 one in 100 12. (1 point) Shoot from the hip first: which one of those would you pick off the bat, without doing any math? GG then supplied the doctors with this, to assist them: This is why so glad there are so many nursing students in my stat classes. Someone has to keep these damn doctors in line with a totally rad name "The probability that a randomly selected woman has breast cancer is about 0.8% (that is, 0.008). If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 9 percent that she will have a falsely positive mammogram. 13. (3 points) OK- now do the math cancer actually has it? What's the probability that a woman who testes positive for breast OK, GG.. how'd the docs do? "In one session, almost half the group of 160 gynecologists responded that the woman's chance of having cancer was nine in 10. Only 21% said that the figure was one in 10-which is the correct answer. That's a worse result than if the doctors had been answering at random. See? This is why I'm excited you're here in MTH 243. . . Project Conditional (non independent) Probability COMID's been pushing us around for (as write this just about 2 years. And I know that we're all sick as hell of hearing about but it's a great application of a really, really neat idea in probability. Let's paint yaa piure to motivate it Suppose you got to get tested for COVID, and the test comes back positive 1. (2 points) What is the chance, do you think that you have it? the a scale of 0 to 100%, "" meaning "There's no way have it to "100%" meaning "absolutely have you're just estimating here, based on your intuition, we'll be working out the numbers soon! No anowers are incorrect Also write a sentence or two about why you answered the way you did Now, in perfect world, if you tested positive for something that would mean you had it, without a doubt. Conversely, testing negative for something would mean you wouldn't have. But we don't live in a perfect world we live in a world where tests are sometimes wrong-and tests ike these can be wrong in two different ways. But, they can also be right in two different ways Confused? Totally understandable. We walk through the posibilities here in a chart De you have COVAD Yes To CIS CS scaled as You' Positive that e COVE but the set tut Youtu COVID and comes back.. You COVE the AW COVD, and Negative data you th So, again in an ideal world, you'd want thour green boxes to happen all the time (depending on whether or not you have COVID) and you'd want the red boxes to happen wel, never but that's the problem: those false positive and negatives de tuppent "OK, Sean-what's the big deal? You might ask "So what you get a faise positive test result? The best comes back positive, it was wrong, and maybe you were a little inconvenienced because you had to-quarantine. Big deal!" 2 (2) What about a false negative COVIO test for would argue, a fale negative test for any infectious disease. Why can that be a huge deal? So, yeah, its a big deal to get the test right-but "getting it right means, actually, 4 different things (as we just saw Maxoming the chances of true positives, Maximizing the chances of tru Minimizing the chances of fase positives, and Minimizing the chances of fale negatives nghofTH 25 may not of the So, that's what we study in this question the mathematics of how some of the currently available COVID work! Now, part of what makes this question interesting and not at all trvabis that no one really know for sure, what the prevalence rate for COVID & in the hour or wo of my searching of various CDC sites (among different locales both national and international, I found that it appears as though the rate might average around 1%. So, what figure! So now, let's break the US down into say, mal groups of 100,000 people who are unfermly, randomly distributed). Assuming that indeed, the infection rate is 1N, that means that 1000 of them will get COMID, and the other 93000 won't Let's redraw that table from above with this new information De you have COVAD Yes 1000 OK! That's coming along nicely. No 99000 Maximing the chances of true positive, Maximizing the chances of true negatives, Minimizing the chances of false positives, and Minimizing the chances of false negatives in the of MTH 205 who muy gosh e treatment of the ab So, that's what we'll study in this question: the mathematics of how some of the currently available COVID tests Now, part of what makes this question interesting (and not at all trivia is that no one really know for sure, what the prevalence rate for COVID is in the hour or so of my searching of various CDC sites (among different locales- both national and internationall, I found that it appears as though the rate might average and 1%. So, we'l use the figure! So now, let's break the US down into, say, small groups of 100,000 people who are uniformly, randomly distributed) Ausuming that, indeed, the infection rate is 1%, that means that 1000 of them will get COVD, and the other 99000 won't Let's redraw that table from aboww with this new information! De you have COVAD Yes No 1000 99000 O That's coming along nicely But remember-all 100,000 of those people take a COVID test, some will come back positive, and the rest wil come back negative. And, in an ideal testing world, all the folks with COVID would get positive test results, and all the folks without it would get negative test results Do you have COMID Yes No Positive 1000 0 You take perfect COVID test in thi ideal w and it comes back... Negative 99000 country's COV can't enough to the 25% as an average infectionate as the up in "hot", and other places have vay lower numbers as type this actually On doing well Well with that mean, other than the fact that we're talking about a deadly panders. The math though But, the tests aren't perfect, and we don't live in an ideal world so we need to deal with false positives and false negatives I took a trip back out to the interwets and found that the faise positive rate for the most effective COVID test (the painful-sounding-but-not-too-awful "deep nose saab one was between 0.8% and 4% ers assume, for purposes of illustration, it's 2% (inds in the middle of that rangel So what that means is this: 2% of people who don't have COMID will test positive. And that means the chart would look more like this: Do you have COVID? Yes No 1980 COM 3 of 7 Of a talking about a deady sederic The huko god, But, the tests anew't perfect, and we don't live in an ideal world so we need to deal with tabe positives and tale negative took a trip back out to the interwebs and found that the false positive rate for the most effective COVID test (the painful-sounding-but-not-too-awfuldeep nose s one was between 0.8% and 4%-se's atsume, for purposes of illustration, 2% (nda in the middle of that range So what that means is this: 2% of people who don't here COMID will still best positive. And that means the chart would look more like this De you have COVO Yes No Positive 1980 (2% of that 99000) You take COVIO test and with the in member 100 it comes back... Negative 97020 (the other 98%) So, that's a wild-even with just a 2% fabe positive rate, you have amat 2000 people testing pouvel but that kind makes sense, too: most people don't have COVID. thankfully Ox-now it's time to deal with false negatives Remember, these are the ones we should really be neu about! According to more sites I was able to track down, the false negative rate is ay between 2% and 37% That 37% threw me for a loop-wicked hight But, it's an outier compared to the other one, so let's fa positive rate that's more toward the 2% say 5%. Then, we can extend our last chart to look th De you have COV Yes No Positive 950 1980 You take a COVID text and it comes back. Negative 50 97020 www..com/22 befor DANIND OK! Now we can look again at your answer from up there where you estimated your chance of having COVIO, assuming you tested postwel. We're going to figure out that chance right now, based on the above chart! Chance you have COVID, assuming you tested positive those who have COVID and tested positive all those who tested positive -32% 950-1980 Chance you don't have COVID, assuming you tested positive 1-Chance you don't have COVID, you tested positive - 1-0.32= 68% If tested positive for something like to be sure at least, mostly sh Alas, this is the reality of disease testing even though the percentages of fate positives and false negatives sound "prod" 12% and 56, respectively, they're not applied equally since, thankfully, most people don't have COVIDI fale pot OK! Now we can look again at your answer from #1 up there (where you estimated your chance of having COVID, assuming you tested positive). We're going to figure out that chance right now, based on the above chart! Chance you have COVID, assuming you tested positive those who have COVID and tested positive all those who tested positive 950 950-1980 32% Chance you don't have COVID, assuming you 1-Chance you don't have COVID, assuming 1-0.32 - 68% you tested positive tested positive If I tested positive for something, I'd like to be sure (at least, mostly sure) that I had it! Alas, this is the reality of disease testing even though the percentages of false positives and false negatives sound "good" (2% and 5%, respectively), they're not applied equally (since, thankfully, most people don't have COVID) But, remember..false positives, while annoying, aren't necessarily the ones I'm worried about. No..I'm more concerned about the false negatives (especially after reading what y'all put up here in #2). 3. (2 points)(w) Using the numbers in the above chart (and a similar approach as I did right below that chart), find the chance that, if someone tests negative for COVID, they don't have it. 4. (1 point) That last value was the chance of a true negative from a COVID test. What's the chance of a false negative? OK-84 is a small probability, which is good! But, still.. those people can then go out and wander around, potentially infecting those around them But, also important that that is the fact that we used "average case scenario" false positive and false negative rates. Remember from above that false positive rates can be as high as 4% and false negative rates can be as high as 37% Use these new "worst case scenario" values (let's use an infection rate of 2% for now) and find: 5. (3 points)(w) The chance that someone who tests positive actually has COMID you like, you can use this bansy dandy little tool built for you it'll get you the table like above; all you have to do is show me the fractions you create and give me the final percentagel 6 (3 points) (w) The chance that someone who tests negative actually doesn't have COVID Whoa! Now, a positive test means you're actually about 3 times more likely to not have COVID (while a negative test still means you're pretty likely to not have it, either). Woof-with those false positive and false negative values, it's really not even worth taking a test, honestly But remember! We assumed a disease prevalence rate of around 2%, but no one really knows how prevalent it is (nor how absolutely reliable the tests are). Of course, typing this about a month before you actually turs this in. But have the @ This kind of mathematics is one of the most wondrous examples of truly useful conditional probabilities (we looked at a few many weeks ago in your first project). Conditional probabilities are the cousins of independent ones with independent probabilities, one event's results was not influenced in any way by another; in conditional probabilities, the events are linked. By what, we might not know-but they're definitely not independent. Let's turn away from COVID now, and look at some examples where rates are a bit more solidified if you've ever spent time at lovely East Lake in Oregon, you might know about the mercury levels of some of the fish there". Environmentalists have developed a test for determining when the mercury level in fish is above "permissible levels" set by the FDA and EPA if the sampled fish's mercury level exceeds this permissible level, the test is 99% effective in determining this (which means that, 1% of the time, it will miss the mercury a "false negative"), However, if the fish's mercury level is within permissible limits, the best will say the fish is "OK" 96% of the time (meaning that, 4% of the time, the fish will get a "false positive"] Now, we also have to have a "prevalence of mercury number; I searched high and low for this, but came up empty, so I'll settle on 50% for starters (and scientists can refine that as they go). This kind of mathematics is one of the most wondrous examples of truly unetul conditional probabilities we looked at a few many weeks ago in your first project) Conditional probabilities are the cousins of independent ones with independent probabilities, one event's results was not influenced in any way by another, in conditional probabilities, the events are linked. By what we might not know-but they're definitely not independent Let's taway from COVID now, and look at some examples where rates are a bit more solidified If you've ever spent time at lovely East Lake in Oregon, you mighe know about the mercury levels of some of the fish there". Environmentalists have developed a test for determining when the mercury level in fish is above "permissible levels" set by the FDA and EPA, if the sampled fish's mercury level exceeds this permissible level, the test is 99% effective in determining this which means that, 1% of the time it will miss the mercury.a "Tale negative"). However, if the fish's mercury levels within permissible limits, the test will say the fish is "O 96% of the time (meaning that, 4% of the time, the fish will get a "faise positive) Now, we also have to have a "prevalence of mercury number, I searched high and low for this, but came up empty, soll set on 50% for starters (and scientists can refine that as they go If a fish tests positive for mercury, what is the probability that the fish's mercury level exceeds permisible levels? And what if the fish tests negative what's the chance's clean Ah, what the hello a little coding in Excel, shall we? Rad! So now, you have a template for running the numbers on any test! Lets give it a shet! (no need to show any spreadsheet stuff/work for these tasume you're doing them the way you did the ones above There is a test for Down's syndrome that can be done during the 4 month of pregnancy for "tik" mothers an at-risk fetus actually has (or will develop) Down's syndrome, the test is accurate 65% of the time meaning it has a 35% fale negative rate), but also carries a 7.2% fase positive rate. A pregnant friend of mine took the test, and the test came back positive (per the CDC, the incidence of Downs in the general population is 1 in 500 for her age group; it increases super-exponentially as the mother gets older) 7. (2 points) What is the chance that her baby will have Down's syndrome? &(2 points) How about the other way what if she tested negative? What would have been the chance her baby wouldn't have had Downs? When I shared these results with her, she felt more at ease. But, then she also asked me a great question.one that I had asked myself many times "Why the hell would you even do this test , no matter what the test says, there's a 99.9% chance of no Downy?" This is called "s a "Bayer Theo Welp, this is why these kinds of mathematical analyses are great there are three different things going on in each! The sensitivity of the test (that's a fancy word that medical folks sometimes use to mean "true positive The specificity of the test byen, you guessed-the "true negative rate) 50, looking above, that 65% accurate test at Ding Downs will be the sensitivity (the "35%" sabe negative because the fetus is at risk of developing Downs (that's the condition: .65% of the time, the test will say sobre positive") The other 33% of the time, it won't so it's abely negative" That test's specificity would be 52.8%, for the same reason of the fetus is not at risk of developing Downs 922% of the time, it will say "no Down The other 7.2%, it was falsely positively result Now, thoor numbers Sugged me, so I decided to look at some updated values found this study, which used a test with updated sensitivity and specificity values sensitivity: 97.25%, speccy 95. Let's use those! 91 point) What's the false positive rate for this new test? 10. (1point) What's the fale negative rate for this new? 11. (2 points) Repeat #7 using this new test (same incidence of Downs as up there Welp, this is why these kinds of mathematical analyses are great: there are three different things going on in each! The rate of whatever you're testing for. The sensitivity of the test (that's a fancy word that medical folks sometimes use to mean "true positive rate") The specificity of the test (yep, you guessed it-the "true negative rate") So, looking above, that "65% accurate test at ID-ing Downs" will be the sensitivity (the "35%" is a false negative because, if the fetus is at risk of developing Downs (that's the condition): . 65% of the time, the test will say so ("true positive"); The other 35% of the time, it won't (so it's "falsely negative") That test's specificity would be 92.8%, for the same reason: if the fetus is not at risk of developing Downs: . 92.8% of the time, it will say "no Downs" ("true negative"); The other 7.2%, it was falsely positively result. Now, those numbers bugged me, so I decided to look at some updated values. I found this study, which used a test with updated sensitivity and specificity values (sensitivity: 97.2%, specificity: 99.8%). Let's use those! 9. (1 point) What's the false positive rate for this new test? 10. (1 point) What's the false negative rate for this new test? 11. (2 points) Repeat #7 using this new test (same incidence of Downs as up there). See the difference a test can make? Now you're beginning to see why so many folks argue about COVID testing: it really depends on which test! If these kinds of analyses are making your head spin, you're not alone. Even doctors screw this stuff up Gerd Gigerenzer, a cognitive psychologist at the Max Planck Institute for Human Development in Berlin, asked doctors in Germany and the United States to estimate the probability that a woman in the general population with a positive mammogram actually has breast cancer. Here's the language he used: "A 50-year-old woman, no symptoms, participates in routine mammography screening. She tests positive, is alarmed, and wants to know from you whether she has breast cancer for certain or what the chances are. Apart from the screening results, you know nothing else about this woman. What percent of women who test positive actually have breast cancer? What is the best answer? nine in 10 eight in 10. one in 10 one in 100 12. (1 point) Shoot from the hip first: which one of those would you pick off the bat, without doing any math? GG then supplied the doctors with this, to assist them: This is why so glad there are so many nursing students in my stat classes. Someone has to keep these damn doctors in line with a totally rad name "The probability that a randomly selected woman has breast cancer is about 0.8% (that is, 0.008). If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 9 percent that she will have a falsely positive mammogram. 13. (3 points) OK- now do the math cancer actually has it? What's the probability that a woman who testes positive for breast OK, GG.. how'd the docs do? "In one session, almost half the group of 160 gynecologists responded that the woman's chance of having cancer was nine in 10. Only 21% said that the figure was one in 10-which is the correct answer. That's a worse result than if the doctors had been answering at random. See? This is why I'm excited you're here in MTH 243. . . Project Conditional (non independent) Probability COMID's been pushing us around for (as write this just about 2 years. And I know that we're all sick as hell of hearing about but it's a great application of a really, really neat idea in probability. Let's paint yaa piure to motivate it Suppose you got to get tested for COVID, and the test comes back positive 1. (2 points) What is the chance, do you think that you have it? the a scale of 0 to 100%, "" meaning "There's no way have it to "100%" meaning "absolutely have you're just estimating here, based on your intuition, we'll be working out the numbers soon! No anowers are incorrect Also write a sentence or two about why you answered the way you did Now, in perfect world, if you tested positive for something that would mean you had it, without a doubt. Conversely, testing negative for something would mean you wouldn't have. But we don't live in a perfect world we live in a world where tests are sometimes wrong-and tests ike these can be wrong in two different ways. But, they can also be right in two different ways Confused? Totally understandable. We walk through the posibilities here in a chart De you have COVAD Yes To CIS CS scaled as You' Positive that e COVE but the set tut Youtu COVID and comes back.. You COVE the AW COVD, and Negative data you th So, again in an ideal world, you'd want thour green boxes to happen all the time (depending on whether or not you have COVID) and you'd want the red boxes to happen wel, never but that's the problem: those false positive and negatives de tuppent "OK, Sean-what's the big deal? You might ask "So what you get a faise positive test result? The best comes back positive, it was wrong, and maybe you were a little inconvenienced because you had to-quarantine. Big deal!" 2 (2) What about a false negative COVIO test for would argue, a fale negative test for any infectious disease. Why can that be a huge deal? So, yeah, its a big deal to get the test right-but "getting it right means, actually, 4 different things (as we just saw Maxoming the chances of true positives, Maximizing the chances of tru Minimizing the chances of fase positives, and Minimizing the chances of fale negatives nghofTH 25 may not of the So, that's what we study in this question the mathematics of how some of the currently available COVID work! Now, part of what makes this question interesting and not at all trvabis that no one really know for sure, what the prevalence rate for COVID & in the hour or wo of my searching of various CDC sites (among different locales both national and international, I found that it appears as though the rate might average around 1%. So, what figure! So now, let's break the US down into say, mal groups of 100,000 people who are unfermly, randomly distributed). Assuming that indeed, the infection rate is 1N, that means that 1000 of them will get COMID, and the other 93000 won't Let's redraw that table from above with this new information De you have COVAD Yes 1000 OK! That's coming along nicely. No 99000 Maximing the chances of true positive, Maximizing the chances of true negatives, Minimizing the chances of false positives, and Minimizing the chances of false negatives in the of MTH 205 who muy gosh e treatment of the ab So, that's what we'll study in this question: the mathematics of how some of the currently available COVID tests Now, part of what makes this question interesting (and not at all trivia is that no one really know for sure, what the prevalence rate for COVID is in the hour or so of my searching of various CDC sites (among different locales- both national and internationall, I found that it appears as though the rate might average and 1%. So, we'l use the figure! So now, let's break the US down into, say, small groups of 100,000 people who are uniformly, randomly distributed) Ausuming that, indeed, the infection rate is 1%, that means that 1000 of them will get COVD, and the other 99000 won't Let's redraw that table from aboww with this new information! De you have COVAD Yes No 1000 99000 O That's coming along nicely But remember-all 100,000 of those people take a COVID test, some will come back positive, and the rest wil come back negative. And, in an ideal testing world, all the folks with COVID would get positive test results, and all the folks without it would get negative test results Do you have COMID Yes No Positive 1000 0 You take perfect COVID test in thi ideal w and it comes back... Negative 99000 country's COV can't enough to the 25% as an average infectionate as the up in "hot", and other places have vay lower numbers as type this actually On doing well Well with that mean, other than the fact that we're talking about a deadly panders. The math though But, the tests aren't perfect, and we don't live in an ideal world so we need to deal with false positives and false negatives I took a trip back out to the interwets and found that the faise positive rate for the most effective COVID test (the painful-sounding-but-not-too-awful "deep nose saab one was between 0.8% and 4% ers assume, for purposes of illustration, it's 2% (inds in the middle of that rangel So what that means is this: 2% of people who don't have COMID will test positive. And that means the chart would look more like this: Do you have COVID? Yes No 1980 COM 3 of 7 Of a talking about a deady sederic The huko god, But, the tests anew't perfect, and we don't live in an ideal world so we need to deal with tabe positives and tale negative took a trip back out to the interwebs and found that the false positive rate for the most effective COVID test (the painful-sounding-but-not-too-awfuldeep nose s one was between 0.8% and 4%-se's atsume, for purposes of illustration, 2% (nda in the middle of that range So what that means is this: 2% of people who don't here COMID will still best positive. And that means the chart would look more like this De you have COVO Yes No Positive 1980 (2% of that 99000) You take COVIO test and with the in member 100 it comes back... Negative 97020 (the other 98%) So, that's a wild-even with just a 2% fabe positive rate, you have amat 2000 people testing pouvel but that kind makes sense, too: most people don't have COVID. thankfully Ox-now it's time to deal with false negatives Remember, these are the ones we should really be neu about! According