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