 Welcome back to our last review for an exam in Math 1030, Contemporary Mathematics for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misseldine. Now, of course, when I say the last exam, I'm referring to exam four here. It's not a final exam for Math 1030. We actually don't have a final exam, not a comprehensive final exam, at least in the traditional sense. This is the last exam, so do make a distinction there. The last exam versus the final exam. This exam will be very much in the same structure as the previous three exams we've seen in this course. There's probably not a lot of new information about the structure of the exam, and as such, we're going to mostly focus on the topics that will be covered on the exam. The only structural difference I can mention here is the number of questions here. The multiple choice section will be a little bit longer on this exam. Each question will still be worth five points each, but there's going to be 12 multiple choice questions, thus making the multiple choice section worth 60 points each. And then as a consequence, the free response section will be a little bit shorter, only four questions there. So that gives us a grand total of 16 questions on this exam. So this exam is going to cover the topics from lesson 29 all the way to the end, which is lesson 36. And so the main topics that you're gonna see on this exam is the topics of probability and statistics, okay? And so there's gonna be some counting, there's gonna be some probabilities, there's gonna be some experimental design, and as such, you'd be expected to answer some of the questions related to these type of topics. So without further ado, let's talk about the specific types of things that you should be seeing on this exam. What should you be studying as you prepare for exam four? So let's begin with the multiple choice section as usual. So this very first question that you're gonna see on the multiple choice is gonna be something taken directly from homework 29. So this is in a lesson 29, we're introduced to the notions of sets, so things like set A, set B. In this context, we're just gonna think of just those collections of numbers or objects and things like that. Our motivation for introducing sets is so that we could talk about sample spaces, events, that is we use sets to calculate probabilities, but just from the notational perspective, I want us to understand the basic symbols that are involved with set theory. So if you're given some number of sets, so in this case, you're given two sets, maybe you're given three sets, probably not more than three. I'm gonna be asking you to do some calculations like can you take the union of a set? Can you take the intersection of a set? Things like that. Can we find the complement of a set? So like what's the intersection of a intersect, b complement? Now of course, if I give you a complement, then we do need to describe what is the universe of the set that they're living in. That's a possibility that could happen with complements come into play. So be prepared to do these basic calculations of sets, unions, intersections, complements, and then I'm gonna ask you to count the cardinality of the set. That is remember the symbol here asks you to determine how many elements are in the set. That makes it nice and easy for a multiple choice question. For us when you compare these things together, take the union, I don't need you to write the whole set. I just wanna know how many elements are gonna be in it. So if you think there's 12 elements, then you'd say the answer is C. Okay, so this will be similar to the type of things we did in lesson 29 and it's a company assignment there. Question number two, this question could take on a couple of different forms, but essentially what it's gonna do is it's gonna ask us to use the fundamental counting principle. The fundamental counting principle told us, remember, if we have independent events, then the product of the number of choices each event will tell us how many things we have together. So in this case, we have something like, okay, there's 25 students on one floor, 20 in on another, 18 on another, 21 there, how many different committees we can make. These are some various counting principles. Counting principles came up in lessons 30 and 31, had to do with permutations and combinations and things like that. So question number two is gonna be one of these counting principle type problems. I want us to understand and show our understanding of these counting techniques here. Question number three is gonna be a question about some probability. I should say more statistics because this is more of an enumeration question. So really the type of things we saw in lesson 34 is what I should say. That is, we're gonna try to count the number of objects in some type of collection of some kind. So this could be like the one estimate sampling that we've talked about, the one sample estimation, I should say, or like the two sample estimations. Can we predict like how large this thing is going to be? In this example, we have 200 coins, some are pennies, some are nickels. We take a handful and we have a sample. So can we guess then how many coins are in the jar? So using some sampling techniques to find proportions that then help us estimate the size of the population. These enumeration techniques are exactly what we did in lesson 34 and you should be prepared to do something like that. Question number four here, I really just want us to make sure we understand the notation that's used in these counting problems. So things like permutations and combinations and factorials, can you make a calculation or can you compute something involving these symbols? Permutations of course were introduced in lesson 30, combinations in 31, factorials have been of course throughout this entire unit. Just be able to use, to be able to understand these calculations and work through them. That's what I wanna see with question number four. Fairly straightforward there. Question number five is gonna be a question about combinations. So as opposed to that previous question, which was just, I want to see some general counting, I'm not gonna give you specific details. I can tell you that question number five is gonna be specifically involving combinations of some kind. So you might wanna go back to assignment 31 to see a little bit more about that if you need some extra practice. Question number six is gonna be a question about sampling. So some type of sampling is gonna be described to you. So like a survey was conducted and it's gonna tell you how the participants in the survey were sampled. What type of sampling is it? Is it cluster sampling, stratified, code sampling, random sampling? I want you to be able to correctly identify the sampling technique that's used and those were all defined in lesson 35. Let's move to the next page of the multiple choice section. Question number seven is gonna be a question about permutations. So question five, we just had a question about combinations. Remember combinations are unordered list as opposed to permutations which are ordered list. And we talked about those in lesson 30. There's lots of different types of permutations we could learn about. This question number seven can ask you any of them. So do make sure you feel comfortable working with combinations and permutations and the other types of counting techniques that we've seen in those lessons. Another example of course comes down to question number eight where we had things like the inclusion-exclusion principle which helps us be able to count unions of things because sometimes the event, the outcomes are not always mutually exclusive. I should say the events are not mutually exclusive. So how do you compute the cardinality of a union? The basic principle is well the union's cardinality is the cardinality of the two events added together but you have to subtract their intersection like so because you've counted the thing that that shows up in the middle twice. So you have to cancel that thing out. And so therefore with question number eight think of it as like question number one but a little bit more advanced. So you'll have things like unions and intersections, compliments perhaps. And I wanted to see you be able to count the sets probably a union using inclusion-exclusion. Use these set techniques to try to evaluate what's the cardinality of the set because when a probability is equally likely, the events I should say are equally likely then computing the probability comes down to finding the cardinality of these sets. So computing cardinality is a very important thing. We introduced that in lesson 29 I've used this throughout the entire unit. Be prepared to do something like that in question number eight. Question number nine and 10 are gonna be some probability questions properly. So question number nine you can expect that some type of probability is being described and I want you to find the probability of the event. So like the question you see on the screen right here a student is guessing on a math quiz what's the probability that they're gonna ace the quiz that they just guessed their way through it. So can we calculate the probability of a specific event? Question number 12 is gonna have to do with probability models, right? Remember a probability model is a function which takes on only non-negative values such that when you take the sum of all of the outcomes in the sample space you end up with a probability of one, that is 100%. So question number 10 is gonna ask you given some information about a potential probability function how do you finish the function in order to make it a probability model? So both of these topics, the probability of an event likewise the probability model these were introduced in lesson 32 which gave us all the principles of probability for which we then took the counting principles we learned about in the previous lessons 29, 30 and 31 and then applied them to these probability models and thus we're able to calculate some probabilities of various outcomes and events. All right, we haven't done too much about statistics yet other than like a sampling question I think. Yeah, so let's do some let's do more on the stats sides here. I guess enumeration, I'm trying to do some sample proportions, that's a statistic question but okay, let's do some more statistics questions. Now question number 11 is gonna be one of those statistics questions for which again, a lot of it's gonna come down to some vocabulary when we learned about statistical design and how do you run a clinical trial? There's a ton of vocabulary there that we have to be aware of. Question 11 is gonna ask us about that. So for example, question 11 here is gonna describe a potential trial that can be conducted to collect data. Is this a single blind trial? Is it a double blind trial? Or maybe it's neither of those things. So do we understand the difference between these type of things? Also, if I describe a clinical trial could we identify which of the groups is the control group? Which of the groups is the treatment group? What is the treatment itself? So question 11 is gonna test our understanding of the terminology with regard to clinical trials and statistical design. And so then that these are all terms that were introduced in lesson 36. You're gonna wanna know what those terms mean so that when asked, you can correctly identify, oh, this is a double blind trial or whatever it is, okay? Then question number 12 is the last question, the multiple choice section. This is gonna be another question about computing the probability of an event. But this one's gonna be a little bit more challenging than we saw on question nine. Question nine, the event will be fairly straightforward and just utilizes our understanding of probability. Question 12 though, it's gonna start to employ some of the counting techniques that we've been using to a probabilistic situation. So to a random variable. So like in this situation, if we have our standard deck of cards, what's the probability that we'll draw certain cards, right? So there's some counting principles come into play then you apply principles of probability and then you can calculate what's the probability of drawing a red card or a diamond, something like that. So OR has to do with like unions. So maybe inclusion, exclusion comes into play there, but maybe the events are independent. So number 12 definitely is a culmination of all of these principles we've learned about with counting and with probability. And that'll be the last question in the multiple choice section. So let's move on to the free response, which is much shorter this time because the multiple choice section because of the probability of statistic questions is a little bit longer than usual. So question number 13, this one's only gonna be worth eight points. Remember all of the multiple choice questions were worth five. 12 times five gave me 60 points there. Question number 13 is an eight point free response question for which some type of random variable is gonna be described to you. Like in this case, we have a spinner board with four different colors. And so if we were to spin the arrow twice, what is the possible sample space there? And so question number 13 here, I want you to describe the sample space of some type of random variable that will be described. And so you would write it as a set. So this would be something like, oh, the sample space would be like green, green. The same, it would also include the green, blue, you know, you'd actually write it out. What are all of the possible outcomes of this random variable that's described here? So this of course was introduced to us in section 32. So feel free to go back to that lesson if you need some more practice. And of course the company assignment as well. Question number 14, this is gonna be a question, and this is their last question from the probability chapter here. This is gonna be a question about expected value. So in this question, you're gonna have some random variable explained to you. This particular one has to do with some street vendor playing some type of marble game. And you can win some money by drawing marbles out of a jar or things like that. So there's just a random variable and you're expected to compute the expected value of that random variable. This is the entire topic that was covered in lesson 33 and honestly nothing from lesson 33 has been described thus far. So by all means, question number 14, which is worth 10 points, will give you an opportunity to compute the expected value of a random variable. So the expected value is kind of like the average, the mean. It tells you what should you expect if you were to play this game over and over again on average, what would you expect to be your winnings from playing such a game? All right, so now we're to the last page of this exam. We're blowing through this one right faster than we usually do. But again, we understand how these exams work. We just needed to know about what are the topics, what are the types of questions we're gonna be asked here. So we can get through this a lot faster and we're doing pretty good. We're in a much better place this time of the semester than we probably were much earlier. Questions 15 and 16 will both be about statistics kind of formatted in a similar way to the multiple choice question that we saw earlier for which you're gonna be described some type of clinical trial. So we have things about some smell and taste treatment. How does that work out? How does aspirin, you know, when people take aspirin, how does that affect things? So you're describing some type of statistical study and then you have to make some analysis of it. So like, can you, and number 15, can you describe who the target population is for this study and who was the sample for the study? Again, put this in words. So you would actually like write out a sentence or two to answer these questions or in this case, describe them. Could you describe like, I could ask about the placebo effect. So in words, you would describe how would the placebo effect manifest itself in this situation? Question number 16 kind of similar. You're describing some type of clinical trial and then it's like identify any compounding variables that could interfere with the study. Could you describe any type of sampling bias that might exist in the study there? So question 15, it's gonna focus more on the mechanical sides of the statistics, like what's the anatomy of the statistics? What's the population, the target population? What's the sample? How has the study been framed? What's the placebo? Is, you know, who's the control group? Who's the treatment group? That type of stuff. Again, really much question 15 is gonna ask about the anatomy of your statistical study there. While 16 is then gonna look at the limitations of that study. Well, that's what we're gonna talk about. Like what variables did we not control for? Are there any biases inside of that? Like again, exploring the limitations of these statistical studies. All right, both of these questions, 15 and 16 are worth 10 points. And unlike the previous questions, I'm not expecting a calculation of any kind. I'm actually expecting a writing sample from you. So you would write the answer. Again, a couple sentences, a small paragraph should be able to answer these things and do make sure that you thoroughly answer the questions with regard to these statistical design questions. All right then. And so that gets us to question number 16 and which is the last question on this exam. This is the last exam for the semester. So fantastic for making it this far. If you do have any questions for exam four that haven't already been answered in this video, I would encourage you one, take a look at the exam syllabus as there's a lot of useful information there that of course is pertinent for this exam. Look at the other study resources that you can find on Canvas such as the practice exam. That can be very helpful. But most importantly, if you do have any questions, please reach out to me. I'd be glad to answer any questions that you have. Work together with your classmates and also remember just to endure to the end, right? You know, we're here at the end of the term and this is one of our last chances to get a good mark in the grade book that has a significant weight on our final grade. I know you can do it. It's our last exam, but this is not necessarily the hardest one. It really comes down to some of these counting principles and understanding the strengths and limitations of statistical research. If you understand those things, you're gonna be do just fine on this test. And if you don't, we'll get some help and you'll get there. Everyone I can trust can do super fantastic on this exam. Best of luck.