The following is a solution to number five and this deals with proportions. And the minimum sample size is necessary in order to satisfy certain conditions. So on this first one, we're asked to find to find the minimum sample size necessary to be 80% confident with a margin of error, no more than 5% points. And there are two parts one where we know The P one and P two and the other one where we don't know. So whatever, we don't know, we have to use .5 because that maximizes are in. So the less, you know, the larger end must be, so in equals the Z star, which is the critical value square and that's 1.282. So uh we square that. Now if you don't know how to find that that you can get in a table but you can also get in the calculator. Now, normally I say go ahead and memorize this, these Z scores, but 80% is kind of an uncommon one. I mean it's, You should know it, but just in case you just make the area .8, The mean is zero. The standard deviations one and just make sure it's center, so you want this center here and then you paste and that's where I get that 1.282. Okay, so that's that part. And then in here. So since I don't know what P one and P two are, I'm just gonna save 20.5 and 0.5 for p one and one minus P one and then plus P two is also 20.5 and then one minus 10.5 is also 0.5. And then I divide that by the margin of error square 2.5 squared. And whenever you plug that in you should get Uh once I find it here so 328.7. But then you always always always round up to the next hole number because we can't have in decimals as sample sizes. So it should be 329. Okay the next part is where you actually know what P. One and P two R. And we set it up the same way but we're actually gonna use the values so 1.282 is still disease score. And then here we're gonna say 0.2 times 0.8 so one minus 10.2 is 0.8. And then plus this P two is 20.65 and then one minus 10.65 point +35. And then we divide that By the margin of error which remember was .05 squared. Okay so whenever you plug that in you should get 254 .7 which of course rounds up to 255. So no decimals whenever you're looking for in. Okay. Part two is the 90% confidence level with a margin of error of 0.2. So again P one P two are unknown. So I'm gonna use 20.5 and equals 1.645 is the Z score for 90% And then .5 times five Plus five times 5. Since I don't know anything. And then divide that by .2 squared. And he plugged that in the calculator should get quite a large number here. So 33 82 0.5. And then of course round that up to 33 83. That's the minimum sample size necessary. Mm. Okay. And then let's say you do know what P one and P two are. Well, same thing. We're gonna do 1.645. Since we're still wanting to be 90% confident. And then this time we're actually going to use those numbers. So .75 and then 1 -175 which would be .25 and then plus six three and then one minus +63 is +37. And then we divide that By that standard air of .2 squared. Okay, so you plug that in And should get 28 45.3. Now, we're always talked around down in the you know, in early math button statistics, we don't do that. We always round up. So it's gonna be 28 46. Always round up. Never round off. Even if it's less than than the 5:28 46. Okay. And then the last 1 95% confident. So that means that Z score is 1.96. We're gonna square that. And since we don't know what P one and P two R. I'm gonna use 20.5 point five .5.5. And then we divide that by the air squared which is .10 squared. And then whenever you plug that in you should get 192.0 Zero, something like maybe 02 or something like that. You still round up. I know it seems you know way off. We're taught early on. Okay, well that's just 192. But that's not the case. We round up up up if there's a decimal and also 92.02 goes up to 193. And then the next part is where we actually know what the the P one and P two R. So we still use the 1.96 squared since we still want to be 95% confidence And we're gonna save .11 times .89 Plus 3 7 times 6 3. And then we divide that by that standard air of 0.10 square or the margin of air. And that gives us 127 .16, which of course goes up to 1 28. So these were the necessary, the minimum necessary uh sample sizes. If we wanted to satisfy those following conditions