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Advertisers contract with Internet service providers and searchengines to place ads on websites. They pay a fee based on thenumber of potential customers who click ...

Question

Advertisers contract with Internet service providers and searchengines to place ads on websites. They pay a fee based on thenumber of potential customers who click on their ad. Unfortunately,click fraud—the practice of someone clicking on an ad solely forthe purpose of driving up advertising revenue—has become a problem.According to BusinessWeek, 45 % of advertisers claim they have beena victim of click fraud. Suppose a simple random sample of 290advertisers will be taken to learn more about

Advertisers contract with Internet service providers and search engines to place ads on websites. 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. According to BusinessWeek, 45 % of advertisers claim they have been a victim of click fraud. Suppose a simple random sample of 290 advertisers will be taken to learn more about how they are affected by this practice. Use z-table. a. What is the probability that the sample proportion will be within ±0.05 of the population proportion experiencing click fraud? (to 4 decimals) b. What is the probability that the sample proportion will be greater than 0.52? (to 4 decimals)



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 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

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

Okay, so this problem is about comparing proportions between two different groups. So we're given Ah, one group that watched one commercial and today is the total number of people in that group 150 and sabes, the total number of people who watch Commercial B, which is 200. We're interested in this proportion of people who recall the commercial later, so I'll call this piece of a hack. That's what we observed from the data at 63 guided by 150. So point for two piece of be hat similarly is 60 divided by the total number in that group. So 0.3 Question eight asks us to test the null hypothesis. That piece of a true proportion of people who would recall commercial A is equal to piece of B, and we're told Alfa equals 0.5 So this means is that our critical value is 1.96 So what we'll do is we'll find our test statistics D, and then we reject little hypothesis. If the absolute values census in the two tailed test is greater than 1.96 is greater than the critical value. So how do we find our the test statistic z well for a test like this. Z is given by the formula p a hat minus p be hacked, which we already calculated we already known. Divided by the standard air. A standard air is given by the formula the square root of P times one minus p which I'll be fine in a moment divided by one minus the number of people in one group plus one minus the divide by the number of people in the second group. Now P is the total number of people. Um, if we group, if we treat our groups is one single population, what is the total proportion that has whatever feature were interested in? So in this case that we call the commercial, we just lump it all together. 63 plus 60 is the total number of people that we call of commercial, and the total number of people is 150 plus 200. So this is 0.351 So if I used this and use my ends of A and B that I calculated before than the standard air that you get from the formula is 0.54 So then I can get my test statistic. Z is point for two minus 20.3. Divided by 0.51 is equal to two point 549 Okay, so now we have our test statistic we have our critical value is 1.96 so we can conclude since Z is greater than 1.96 we reject the null hypothesis and say that there is not There is, uh, sufficient evidence to support the claim that the proportions differ. Okay, so that was part of a all right. In part, Beat asks us to find the 95% confidence interval for this difference in proportions for p A minus p b. So since this is the 95% confidence interval, Alphas again 950.5 which means our critical value is again 1.96 The 95% confidence interval is then given by well are observed difference plus or minus the critical value 1.96 times p a hat times one minus p a hat plus PB hat. I was one minus PB hat. This charm is divided by the total number in rue pay in this term's divided by the total number in Group B, so we can just plug in the numbers and we see that this is point for two times one minus point for two, divided by 1 50 plus 0.3 times one minus 10.3 led by 200. Take the square root times that by 1.96 and this number that we would get is 0.1 01 before and this difference is just point for two minus 0.3 just 0.12 So in the end, we have our confidence. Interval its 0.12 minus 0.1014 That's the lower bound. The upper bound is 0.12 plus 0.1014 So the final answer his 0.186 0.2 to 1 for there you go.


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