MEDICAL MATHEMATICS II 2025 JULY END OF SEMESTER EXAM

All right, welcome to this video. Today we'll be looking at the final exam for the July intake 2025, and this is medical mathematics two. Okay, so this paper has three sections. There's section with 10 multiple choice. There's section with structured questions, and there's also section with structured questions. So we're going to solve all the questions in this paper. And by the end of us solving this paper, then you must be ready for the final what? Exam. So we'll start from the multiple choice one, and then we'll move like that. Okay, so the first question says hospital records an average of four postoperative infections per month in surgical ward. Assume the number of infections in in the month follows Poisson distribution with mean lambda is equals to four. What is the probability that at most two infections occur in given month? Okay, so first of all here we've been told that the number of infections in month follows Poisson distribution with this mean. So we're going to say, "Okay, let the number of infections in month be represented by So this here is following Poisson distribution. Okay, with average of what? Four. Okay, so now we know that under the Poisson distribution the PMF function is given by this formula here. Okay, so that will be to the power minus lambda and then lambda to the power divided by factorial. We know that's the PMF. So, meaning if you want to find the probability, we are going to use that. Okay, so now here the question is, what is the probability that at most? So, what does at most means? Okay, at most simply means less or equal to. Okay, less or equal to. So, at most two infections occur in given month simply means less than or equal to two infections occur in month. So, we are trying to find the probability that is less than or equal to what? To two. Okay, so that's the probability there. So, this probability here, you can easily find it by saying, "Okay, let me find the probability when is equals to zero plus the probability when is equals to one plus the probability when is equals to what? To two." Then you just add them like that. That will be your answer there. Okay? So, if you want, you can do them individually, meaning you can find this probability by just using this formula here. Okay, so this will look something like to the power minus lambda, but know the lambda is four. So, that is minus four and then four to the power zero because is now equals to zero over zero factorial plus here now it will be minus four then four to the power one over one factorial plus minus four four to the power two over what? Two factorial. And then you add this. But, the good news is that you don't even have to do this. Okay? You don't even have to do this because there's shorter method where you use the cumulative Poisson table. So, the cumulative Poisson table has probabilities where is less than or equal to certain value of what? Of Okay, let me write there not two. certain value of So, if you want to find the probability, okay, you just have to know the average. In this case, my average is four and then you have to know this value of what? Of So, meaning that without even solving this, can just simply say the probability that is less than or equals to is equals to So, my is two, my average is four. So, just go to the probability distribution table and get the value. Okay. the the Poisson cumulative table. Okay, so when go to the table the table is here. So, this is the Poisson table here. Okay. So, you can see that this this row here has lambda there, which is the average. Okay, and then when you come down here, you you are finding the value of the smaller letter or sometimes they say So, our lambda was four. Okay, our lambda was four and then our was two. Okay, our was two here. So, I'm just going to go from two, I'm going to the right like this. Okay. And then from the four, I'm coming down like this. So, where they are meeting, they are meeting there. So, the probability is simply 0.2381. Then you come and write 0.2381 without doing any of these calculations. Okay, but if you want, you can see still go ahead and use the calculations here. You're still going to get the same answer. So, this is our answer here and when you check on the options, that answer is on So, we'll select as our best answer. Okay, moving forward. Question two says hospital disease control and outbreak response team is on the lookout for potential monkeypox outbreak in the country. Monkeypox outbreak. Okay, according to the studies on this disease hyperplasia Okay, hyperplasia is potential lead to identifying such cases. So, meaning that if they find patient has hyperplasia then they that patient is more likely to have the monkeypox, right? Okay, so now patient is said to experience hyperplasia if body temperature reading is above 40° Suppose the body reading of patient is assumed to be normally distributed. So, meaning is normally distributed, okay, with the mean with the mean mu is equals to 36 and standard deviation two. But, we know that when we are writing here, you write the variance, right? So, the variance sigma squared is equals to what? To four. Because that is just the standard deviation squared. Okay, so this is what have here. Meaning my standard deviation is just what? Two. So, this is sigma here. So, now they're saying if random sample of 350 patients is screened for hyperplasia approximate the number of students who have to be isolated as potential monkeypox disease. So, these people have to be isolated if they have the disease. So, in other words here, to find that number, we just have to calculate first of all the probability. Okay, of people having the disease or or people being having hyperplasia. And then once you have that type probability, we're just going to multiply by the sample size to get the approximate number of people that need to be isolated. Okay, so now this is normal distribution. So, the probability, okay, that is greater than because here they said high blood pressure happens if is above 40°. So the probability that is above 40° like this. Okay, we're going to find it like how? We have to standardize. So this is probability when standardize the get the greater than When standardize this it will be 40 minus the mu. Okay, divided by the standard deviation which is what? Two. So this is what have. So this will be the probability that is greater than So this is 40 minus 36 four divided by two get two. Okay, so that is what have. So now on the distribution table there there are no probabilities for greater than. So meaning have to change this to one minus the probability that is less than what? Two. Okay, and then now we can go to the table and find this probability. Probability that is less than two same same as is less than 2.00. So now when go to my tables here where is my normal distribution table? My normal distribution table is here. So I'm looking for the probability that is less than 2.00. So two if come here in the first column it's there. Then 00 is there. So meaning this and this two they they intersect at that number there. So that is 0.97725. 0.97725. Okay, so we have to subtract now those two numbers in order to find the probability. So one minus 0.97725 What I'm going to get here will just be 0.02275. Okay, so that is my the probability of someone having the what? The hyperplasia. And then now to find this how many people should be isolated, the mean, meaning they have hyperplasia, we're just going to say the expected number of people is equals to the probability of having this disease. Okay, multiplied by the sample size that we have, which is 350. So multiply here, what are we going to get? Okay, so get 7.9625. round off to these to nearest whole number, get eight patients. Okay, so that will be my answer, and when you look at the options, that answer is on option So we're going to pick as our what? Our final answer there. Okay, we'll move. Question three. rapid diagnostic test correctly detects particular infection with probability of success is equals to 0.0 0.20 on any patient. So meaning the probability is constant. clinic tests is equals to 10 patients. Let be the number of positive test results. What is the probability that at least three tests are positive? So here you have to identify first the distribution of right? So we've been taught the probability is constant on any patient, and we've been given number of fixed trials. So meaning that follows binomial distribution with success probability of success 0.2, and then the is what? 10. So now the question is, find the probability that okay? That at least three tests are positive. So, at least simply means greater than or equal to. So, at least like this. So, this is the probability we are finding. Okay? So, we can change this by the complementary. This will be 1 minus the probability that is less than what? Three. So, this will just be 1 minus this probability that is less than three can be the probability that is equals to zero plus probability that is equals to 1 plus probability that is equals to what? Two. Okay? So, if you want, you can get the PMF of binomial distribution and actually calculate this probability, okay? But, we are lucky because we have cumulative binomial table. So, I'm just going to go to the cumulative binomial table and find this. So, now since we're dealing with discrete case, okay? The probability that is less than three is the same as the probability that is less than or equals to what? To two. This is what we can see here. This is just zero, one, and two. Okay? So, can go to the binomial table for the cumulative probabilities and find this. What need is just the probability of success, 0.2 and the number of what? Trials. Like that plus this number two. So, let's go to the binomial table and find out what these probabilities here with these conditions. Okay, so where is the the probabili- the binomial table? Okay, so first of all Okay, first of all, we identify our Our was 0.2, so it's there. So, we are going in this line here, down. Okay? And then my was 10. So, I'm moving up until find my 10. Okay, so this is my 10 here. So, you can see the 10 is here. The is equals to 10. And then there where the should be less than or equals to 2. Okay, so in in this case here, the is just being represented by this So, meaning that is equals to 2 is there. So, I'm going to move to the right in this line and then still come down here. So, you see where I'm meeting? It's there. So, that is 0.377. Is it okay? What is happening? 0.6778. So, 0.6778 and then we subtract. Okay, so 1 minus 0.6778, that is just going to give me 0.3222. And that is our answer on part Okay, so that's how easily we can actually answer that question. Let's look at the next question. Okay, so the next question says random sample of 100 patients, meaning that my is equals to 100, recovering from medical procedure gives 95% confidence interval for the mean recovery time in days as the one that starts from 72 all the way to 78. Find 99 confidence interval for the population mean recovery time based on this information. Okay? So, what I've done here is that they've given us confidence interval at 95% confidence, right? So, the confidence interval is from 72 to 78. Okay, and this is the confidence interval about the mean. So, to find the confidence that interval about the mean, you use this formula. It will be bar, which is the mean, plus or minus Okay? bar plus or minus the critical value at the same confidence interval multiplied by the significant here, multiplied by the standard deviation sigma divided by the root of This is the formula. Okay? For the confidence interval about the mean. Right? Okay. So, want to orient you on what these things mean here. So, the bar is the mean. Okay? This that looks like six here is not six. It's called sigma. It represents the standard deviation. is the sample size, which we've been given as 100 there. And then this is with alpha divided by two, it is just the critical value at the particular confidence. Okay? So, we get that value from this table that I've written down on the side here. So, if it is 95% Okay? The critical value just be positive 90 positive 1.96. If it is 99, it will be 2.576, just like that. Okay, so this is the formula here. So, now this part here when get the standard deviation Okay? Divided by the root of there. This part here that I've circled is called as the standard error. Okay? It is called as the standard error. And then now, when combine this standard error with the critical value, when get this whole thing here together get what we call as the margin of error. Okay? So, this whole thing here we can actually represent it by the what? By the margin of error. So, can actually say confidence interval about the mean is bar plus or minus margin of what? Error. Okay? And then if use the symbols that are in between to be bar minus margin of error {comma} bar plus margin of what? Error. And this here becomes our confidence what? Interval. Okay? Of course, at the particular confidence level. So, here we've been taught that the confidence interval is 72 to 78. So, 72 to what? 70 eight. And then we are asked to find the confidence interval at 99% confidence. So, meaning that if I'm using the formula to be confidence interval at this 99% bar plus or minus the critical value at that confidence interval multiplied root of So, now this part here, this part here, the standard error. It won't change in this scenario here because we are dealing with the same data which has the sample size meaning the standard deviation there is also the same. So, this part is not changing. What will change is the critical value be because for this one it's at 99% for this one it was at what? 95%. So, meaning that for me to find the confidence interval at 99% just have to find the mean. Okay? This guy can get it from the table and also have to find the what? The standard error. So, how am going to find it? use this information that I've been given here. Okay? So, here it simply means this 72 is equals to this. So, bar minus margin of error is equals to 72. And then on the other side, bar plus margin of error is equals to 78. Okay? So, if want to solve for bar or margin of error, can simply say, "Okay, bar is equals to 72 plus margin of error." So, I've taken this guy to the other side. And then let me replace it in this equation. So, where there's bar, put 72 plus margin of error. Plus have another margin of error, and this is equals to what? 78. So, meaning have two margin of error is equals to 78. This 72 goes to the other side. So, this is minus 72. So, two ME is simply equals to six. If divide both side by two, my margin of error is simply equals to what? To three. Okay? So, that's the long method in which you can calculate the margin of error. But there's also shorter method. So, the margin of error just gives you the distance between the lower value of the confidence interval and the upper value of the confidence interval. So, there's no need to do all these things in the exam. I'm just showing you here so that you'll know. But for you to easily get the ME, margin of error, you just say the upper limit minus the lower limit. So, 78 minus 72, you see that it's still giving us what? Three. Okay. So, I've found the margin of error, margin of error, which is equals to three. And remember the margin of error is this thing here, the standard error multiplied by the critical value. So, meaning that what have is critical value multiplied by standard error. Okay, let me just represent it as SE, standard error is equals to Okay? Three. Cuz when multiply this and for this part, get the margin of error. Okay. So, now for this one here, the confidence interval at 95% what is the critical value? The critical value is 1.9 six. So, this will just be 1.96. Standard error is equals to three. So, want to find the standard of error, which is constant. So, standard of error will just be equals to three over 1.9 what? Six. And when divide Okay, let me divide that. Three divided by 1.96. Okay, it gives me approximately 1.5306. So, this is my standard of error. So, can say, "Okay, have the standard of error to use here." Okay. And then what is my mean? My mean, since have my margin of error, can simply use this formula here. So, mean is equals to 72 plus my margin of error, which was three. This gives me 75. So, my mean is equals to 75. And also have my standard error. So, can now calculate the confidence interval at 99% level. just have to find the critical value at that level. Okay, and you can see at 99% the level is there, 2.567. So, let's calculate the standard deviation now. mean the confidence interval now that we have everything we need. Okay. So, I'm just going to say the confidence interval at 99 le- confidence level will just be the mean, 75 plus or minus the score at 99, 2.576. And then this I'm multiplying with standard error, which is what? 1.5306. Okay, so confidence interval just be equals to what are we going to get there? We're going to say it's 75 plus or minus. When multiply this, get roughly 3.94. Okay, you can check on your calculator. So, my confidence interval will just be equals to 75 minus this or 75 plus it, 3.94. Okay, so get your calculators. what are you going to get? Here I'm going to get 70 1.1. Then on the other side, 78.9. Okay, approximately that. So, this is the confidence interval and can we see it there? Yes, we can. We can see it on part there. So, that is the answer there. Okay, that's the answer there. Okay, so what is remaining is to go to the next question. So, all right. So, thank you so much for watching that video. If you'd like to access more videos, okay, to help you in your pre-med journey or your NS journey, you are welcome to register and you can access the full videos at the script-learning.com. Also, if you have any challenges in any of these pre-med courses, biology, physics, chemistry, or mathematics, you are free to actually subscribe with our institute, okay? we're going to help you lot. 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