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Imagine that you work on the data science team at Hooli, one of the world'$ most prominent digital platforms_ The firm is considering a change to their search ...

Question

Imagine that you work on the data science team at Hooli, one of the world'$ most prominent digital platforms_ The firm is considering a change to their search engine website in which visitors will be asked to answer a single "Yes or No" survey question before they get access to the content on the page. Let uS assume that all visitors to this page fall into one of two categories: (1) Random Clicker and (2) Truthful Clicker: There are two possible answers to the survey question: (1)

Imagine that you work on the data science team at Hooli, one of the world'$ most prominent digital platforms_ The firm is considering a change to their search engine website in which visitors will be asked to answer a single "Yes or No" survey question before they get access to the content on the page. Let uS assume that all visitors to this page fall into one of two categories: (1) Random Clicker and (2) Truthful Clicker: There are two possible answers to the survey question: (1) Yes and (2) No. Random Clickers would click either (1) Yes and (2) No with equal probability You have data indicating that the overall fraction of Random Clickers is 0.34. After a trial period, YoU collect the following survey results: 63% responded Yes and 37% responded No. What is your estimate for P(Yes TC), i.e , the probability that a Hooli visitor responds "Yes to the survey question; given that they were Truthful Clicker?



Answers

Advertisers contract with Internet service providers and search engines to place ads on Web sites. They pay a fee based on the number of potential customers who click on their ad. Unfortunately, click fraud- the practice of someone clicking on an ad solely for the purpose of driving up advertising revenue- has become a problem. Forty percent of advertisers claim they have been a victim of click fraud (BusinessWeek, March 13,2006 ). Suppose a simple random sample of 380 advertisers will be taken to learn more about how they are affected by this practice. a. What is the probability that the sample proportion will be within ±.04 of the population proportion experiencing click fraud? b. What is the probability that the sample proportion will be greater than $.45 ?$

All right, this question asks us about distribution with a population proportion of 0.4 that refers to advertisers saying that they have been victims of click fraud. So are a What's the probability that we find a sample proportion within 0.4 in a sample size of 3 80 So to do this, we can rewrite this probability as the probability that we find a sample proportion between 0.36 and 0.44 which we can solve using normal CDF because we know the lower bound we know the upper bound we know the mean and we can calculate the standard error, which is square root of P times one minus p all over end. And this works out to be 0.88 eight five, then for part B. It wants the probability that we find a sample proportion greater than 0.45 So this is already in a form that we can insert into normal CDF so we can just write normal CDF of of our lower bound and we don't have an upper bound, so we pick a very large number. Then we have our means both point for and our standard air still stays the same because our sample size isn't changing and this gives us a result of 0.233

All right here. We're working with sampling distributions, and we have an example in which we have 40% of advertisers who say that they are a victim of click fraud. What we are using is a random sample of 380 advertisers so that we can learn more about this. So in this case, let's first acknowledge what I have written in blue right up top. Let's start to find some of these pieces of information because it'll come in handy as we progress. So let's start with our mean okay, which is equal to P. And we're told that we have 40% of advertisers who believe this, So that would give us a mean of 0.40 Let's find our variance, which is calculated as p Times one minus p over end. So substituting that value for P. We just found to be 0.4 times one minus 0.4 divided by N and we have a random sample of 3 80 so three eighties R n. That gives us a variance which is equal to 0.632 All right, and now our standard deviation is just the square root of that variance. So taking the square root of 0.0 0632 we get a standard deviation which is equal to 0.25 Let's now figure out what our Z value would be following a normal distribution. You can see that formula written here as well. So now we just need to substitute some of these values in. So we have. We'll get that Z is equal to R P hat which we don't know yet. That's what we're trying to find, minus our mean of 0.4 divided by our standard deviation, which we just found with 0.25 There we go. So this is just because we know that it's going to be useful as we move forward. And we know this is following a normal distribution where probability lies between zero and one. All right, moving on to part A. We want to find the probability that our sample proportion is within 0.4 plus or minus of our population proportion. All right, so you can see that this probability is given by what I have written in blue here. So it's the probability that our sample proportion is greater than or equal to p minus 0.4 or less than or equal to P plus 0.4 And we know what P is because we just found this up top so we can rewrite this as the probability that point four minus 0.4 less than or equal to our sample proportion, which is less than or equal to 0.4 plus 0.4 You can see all I did was substitute that value for P in here. Let's go ahead and simplify this even further, which gives us p is equal to We're just gonna subtract these from one another. So we have 0.36 which is less than or equal to our sample proportion, which is less than or equal to 0.44 All right, now we know that we need to find some Z values here. And so this is where we're gonna come back up to what we just found up here for Z, okay? And we're going to plug some of these values in so we have our probability. Probability will be calculated as 0.36 minus 0.4, divided by our standard deviation of 0.25 which is going to be less than or equal to the sample proportion, which we don't yet have. Minus 0.4 all over the standard deviation 0.25 It is less than or equal to 0.44 minus 0.4 all over our standard deviation again, right? Let's simplify once again, and what you can see is this middle piece here that's just Z that's equal to what we found up here. So this is going to turn into the probability of negative 1.6 that was just simplifying our first piece, which is less than or equal to Z less than or equal to a positive 1.6, again found by just simplifying each of these pieces. Now we need to actually find the Z value of this probability is using our normal distribution table. So again, let's further simplify this make a little bit easier, understand? So we're finding the probability that Z is less than or equal to 1.6, minus the probability that Z is less than or equal to a negative 1.6 right. That's just going to give us whatever that value is lying in the middle here. So if we do that and using a normal distribution table, we see that we have 0.9452 for a probability there, minus the probability equal to 0.548 And that gives us a total probability that our sample proportion is within plus or minus 0.4 of our population proportion, which is equal to 0.8904 All right now, let's say we want to find the probability that our sample proportion is greater than 0.45 So to do this, I'm gonna scroll down a little bit. We know that our this probability will be written as probability that R P hat are same proportion being greater than 0.45 And let's simplify this a little bit. Or let's start to substitute, actually, so this can be rewritten as our probability of P hat minus 0.40 all over our standard deviation 0.25 So it's a very similar process that we just did in part A would be greater than 0.45 minus 0.40 all over 0.25 So you can see we have P hat and this is what we want to know. We want that to be greater than the second piece, and I got the 0.45 because that was given to us right in here. All right, let's go ahead and simplify this. This simplifies to be the probability that Z is greater than two, because again, this first pieces just Z we found that up top. It's written the exact same way. So and then our second piece simplifies to two. This can simplify even further. We'll just rewrite it more like as one minus the probability that Z is less than or equal to two. That way, it's a little bit easier to use our normal distribution table. This gives us one minus 0.97725 which simplifies to a probability equal to 0.2275 That being our probability that the sample proportion will be greater than 0.45

Everybody. My name is Colin. Let's go ahead and look at this Snow mobile and environmental clubs problems. So in court they were asked to find the probability that the individual and the first part is a snowmobile owner. So to do that, we're going to look at this table here. We're going to see that the total number of snowmobile owners is to 95 and the total number of respondents was 15 26. So we know that the probability that the respondent was a snowmobile owner is equal to 0.1933 They were going toe some part two of this problem. We're looking for the probability that the person belongs to an environmental organization. Or is this new move you honor? So to do that, what we need to dio is we need to take the probability that the person that the respondent is a member off an environmental club. So 305 Oops, that is 3 50 30 5/15 26 and added to the probability of that person is a snowmobile owner. However, we've gotta be careful here because it would be easy just to say to 95 or 15 26. But we've got to take into account that 16 of those 305 people who belong to an environmental club also own snowmobiles, so we don't want a double count them. So the number that we need here is actually to 79/15 26 and what that does is that gives us that, not the stove ill owners who were not already previously counted. And when you do that calculation, we're going to do 5 84/15 26 or about a probability of 260.383 and the last part of this first problem, as it's asking us for the probability that they have never used a snowmobile, given that they belong to an environmental organization. And so for that, we're going to look at the people who answered Yes, they have never used a snowmobile, which is 212 and the total number here that we're looking for is the total number of people who are a member of an environmental club which is 305 and so that number right there is about 0.6951 and that is the probability that the person has never used a snowmobile, given that they belong to an environmental organization. Now we're asking Part B, whether or not belonging to an environmental organization and owning a stone will be our independent. And the answer here is a resounding no. You would think that maybe because there are so few snow mobile owners that are also environmental club members, that you could maybe go ahead and say they are independent. But in fact they are not, because not all snowmobile. Not all snowmobile owners do not belong to environmental club, and some folks who own snowmobiles do belong to environmental clubs. So there is some overlap. Those two categories are not independent, so now we'll move on the part, see where were asked to find the problem. If we were asked to choose, turn to survey respondents at random, and then we're going to find some probabilities based on assuming a random selection of to survey respondents. So the first part assets the probability that if we choose to respondents at random, what is the problem? They both own a snowmobile. So to solve that, what we're going to do is we're going to look at the probability that we've got to respondents and this first guy his first responded owns a snowmobile. And that probability is Aziz. We calculated in part a Tu 95/15 26. And since this guy is the same, also owns a snowmobile that's 2 95 15 26 and is your calling. We're doing this. We've got to take both of these responses into account. We're going to multiply those two numbers together, and we're going to end up with a final probability that both people who responded in this case own snowmobiles as 20.374 And now part two of Part C asks us the probability that at least one belongs to an environmental organization matter than doing this captive. Probably that one is. And then the added to the probably that to our I'm just going to have to go ahead and do the shortcut here. I'm going to find the probability that no, you long and then what I'm going to do is I'm going to subtract it from one. So I realize now that I wrote that in a little bit of a confusing way. But What we're going to end up doing is if we say that X equals the number of people, the number of respondents who belong. And of course, our options here are 01 or two because we're choosing to response at random. But I'm going to do is take one. It's attract the probability that none of those respondents belong, because that will give me the same you'll remember as the probability that one is a member and the probability that, too, is a member. So to calculate the probability that none of them are, remember what I'm going to do is look at, uh, I'm going to get the probability that the first respondent is not a member and the second respondent is not a member. So 12 21/15 26. And I'm going to multiply those two together, and I'm going to get 20.6402 And so then I'm just going to subtract that number from one. Like I said, I'm going to get to 0.3598 And this right here is the probability that at least one of the respondents


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