5

In # poll taken In Eebruan = 2005, wznt 8 Matrin 0i Cttof Ol { 33 Gollup aked adulu ! Uay Krtr #porui fan 637 Ld {ey #ere usine $ 95X confidence Itenal Wny minimum ...

Question

In # poll taken In Eebruan = 2005, wznt 8 Matrin 0i Cttof Ol { 33 Gollup aked adulu ! Uay Krtr #porui fan 637 Ld {ey #ere usine $ 95X confidence Itenal Wny minimum Lmcie Suooret accomeluh this? ufe nold benrrend t0MOE

In # poll taken In Eebruan = 2005, wznt 8 Matrin 0i Cttof Ol { 33 Gollup aked adulu ! Uay Krtr #porui fan 637 Ld {ey #ere usine $ 95X confidence Itenal Wny minimum Lmcie Suooret accomeluh this? ufe nold benrrend t0 MOE



Answers

Construct the confidence interval for $p_{1}-p_{2}$ for the level of confidence and the data given. (The samples are sufficiently large.) a. $80 \%$ confidence, $$ \begin{array}{l} \mathbf{n}_{1}-300, \hat{p}_{1}-0.255 \\ \approx_{2}-400, \hat{p}_{2}-0.1 Q 3 \end{array} $$ b. $95 \%$ confidence, $$ \mathrm{w}_{1}-3500, \hat{p}_{1}-0.147 $$ $$ \mathrm{w}_{2}-3750, \hat{p}_{2}-0.131 $$

Into this list. So in exercise one we had a pool that yeah, saw 1910 professionals interviewing job applicants And 26% of them. Mhm. Uh So the biggest interview turned out is the applicant did not know or the gnomic efforts to learn about the company. So we are looking at the sample proportion. The there's a proportionate favor the proportional in favor. You're looking at the number the sample size, the number that we are talking about, the number the size of the the whole study. We are looking at the E. Which is the arrow. We are looking at the critical value. All right. So with this one we had 26%. That said that the the biggest 10 of was the applicant. They don't know about the job. So that is 26%. Which is 0.26. So the converse of deaths or one minus the sample portion would become Yeah. Well uh 74%. That is one my necessary .26 which will become their point 74. We have 1910 professionals. The sample size Then we have the Arab which was 30%. So their point 03 than uh critical value. Since the confidence interval It was 95% would have 0.95 -1 -3.95. And that will give a 0.25. So 5% for that critical value. All right. Thanks for the time. This is the end of the lesson. Cool. Mhm

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

So for this question will be using formula who's he squared times P. That modifying one minus P. And divide all that by E squared. So I started with per a. The P value is .81. It is a 95% confidence interval And the e. value is .02. So before we put all the numbers of formula we need to get Z. And we need to see by changing the confidence in our whole Into a Z score. So first you would change to destinations attracted from one and then get the difference to buy the difference way too. And then that's the number you look up on the sea table and you find out it's 1.96 around it. Yes. Now we can put it in the formula so will be 1.96 squared Times .81 times one minus .1. Then you would divide that all by .02 And you would find out that answer is 1004 out of 78 points six. But because that's like You can have one fraction of this person and a sample size you need to round it up to the nearest whole number and you get 1000 479 for this answer. Yeah. Yeah part beef we use the same formula. So p value it's .81 Lives up to 99% confidence interval. E. is .02. So first again we'll need to change considerable into a Z score. Going to find it carried it .005. And then when you look it up to see table is 2.5 gates. I would put it in our formula. Uh huh. You just do the same um interesting process. There's a switch a number And you get 2,561.5. Again we have to round it up to 2562. Yeah. Then for C. r. standard deviation is eight. Very sorry, P value, P value. It's 21. We have a 95% confidence interval. E. Is too. And because we did this in part a for the 95% confidence in the book, It would be the same number which was 1.96. So then it would be 1.96 squared Tires, Tires, 1 -11 Divided by point there one sq and unit 5012.22. He had to round it up for the minimum it would be 5009 hired 8:13 Yeah.

The following is a solution for number three and we have to find the 99.5% confidence and over for the difference of two population means given this data here and we're gonna use the two samples the interval. Because these sample sizes are so large that the sample size is larger than 30, you can use the T or the Z interval. Um In this case I'm just using the Z. Because those are pretty darn large and it's going to convert converge to the T interval anyway, Roughly, you know, with some rounding. So if we go to stat, this is on the IT4, if we go to staten test, it's this ninth option down here. The two samples the interval. So click nine summary stats should be highlighted now where it says sigma, that's actually just gonna be where your sample standard deviations go. Uh That stands for population standard deviation is, we technically don't know it, but again, that sample size makes it to where it really doesn't matter. So the sigma one or the s one was 2.5. The s two is 4.6. The first sample mean was 27.2. The first sample size was 1 30 the second sample mean was 38.8. And the second sample sizes 1 55. And then for sea level, we're gonna write 550.995. And then we calculate And we get this here -12.81 a negative 10.39. So I'm gonna write that so negative 12.81 and negative 10.39. That's our 99.5 Confidence interval. Which is pretty confident. Okay, so then we go to the 95% confidence interval using slightly different data. So I'm going to go back to the two samples e end because that sample size there were both bigger than 30. So we can use the Z interval and the first sigma in this case of the the S. Was 12 3. The second s would be 14 1. That sample size our sample mean for the first sample was 215.5 and that sample size slightly smaller but still big enough 68. And then X-2 bar was to 87.8 and then the end to was 84. And we're finding the 95% confidence interval. Okay? So .95. All right. And then we calculate and we get this so negative 76.5 to negative 68.1. Okay, So I'm gonna write that down. That's our second Confidence interval negative 76 5 To negative 68 points one. Okay? And that is the two samples the interval


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