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Workers in several industries were surveyed to determine the proportion of workers whofeel their industry is understaffed. In the government sector, 37$\%$ of the respondents saidthey were understaffed, in the health care sector 33$\%$ said they were understaffed, and in the education sector 28$\%$ said they were understaffed $(U S A$ Todary, January $11,2010) .$ Sup-pose that 200 workers were surveyed in each industry.$$\begin{array}{l}{\text { a. Construct a } 95 \% \text { confidence interval for the proportion of workers in each of these }} \\ {\text { industries who feel their industry is understaffed. }}\end{array}$$$$\begin{array}{l}{\text { b. Assuming the sample size will be the same in each industry, how large would the }} \\ {\text { sample need to be to ensure that the margin of error is } 05 \text { or less for each of the three }} \\ {\text { confidence intervals? }}\end{array}$$

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a. Government: 0.3031 to 0.4369Health care: 0.2648 to 0.3952Education: 0.2178 to 0.3422b. $n=443$

Intro Stats / AP Statistics

Chapter 8

Interval Estimation

Confidence Intervals

Christa D.

December 11, 2020

please provided the margins of error for the 95%

Temple University

University of North Carolina at Chapel Hill

Piedmont College

Lectures

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once again, welcome to a new problem. This time we're dealing with confidence confidence intervals and the confidence intervals. Looking at his for proportions. So if you have a population of subjects of people, this population can have P which is the same as the population proportion. You know, this is the same as the population proportion. And within that population proportion you have P had which is the same as the sample proportion. And you can always make predictions Using P uh on P hurt for example, if you wanted to I predict the percentage percentage of females. Yeah in a college campus you can use pr to predict P. The problem though is in estimation there is likelihood good of error. So this is the likelihood of making an error. Mhm. In estimation. Therefore, confidence level, illicit trust in the prediction outcomes for both proportions and means. So then you know you have period plus minus E. We where he is the margin of error and then the E will be the same as Z alpha over to Body called P Hut 1 1 Speed Hut, Oliver, N ZF of two for 95% confidence will mean that this middle parties for 95 And then each one of the Tales is .025. That's each one of the tails. And so z off over 24 95% confidence is 1.96. Of course you have two sides. This would be the negative side in this positive side. 1296. And the reason for that is because we have a plus and a minus. So coming back to these problems, we have surveys of employees. Yeah. In multiple industries uh revealed the uh type of industries which are under stuffed. So the type of industries which are understaffed. And we're seeing that for example, in the government sector, the government sector, the understaffing dessert 37% based on respondents information. And then also we have in the health sector uh 33 under stuffed. And this again is based on responding to information. And and then we have understaffing in the education sector where the level of Under staff in his 28 and the stuffed. Mhm. And all this is based on 2010 U. S. A. Today. Yeah, I think perhaps assume assume 200 walkers Or salvage. So we're making an assumption that 200 workers, we solve it. And then but they were going to say uh determined become in 95 confidence interval mm mm For proportion in order of walkers by industry. Mhm. Uh Under stuffed. And then the body was saying, mhm assuming similar similar sample sizes by industry. Yeah, determined the size. Oh, this sample that guarantees guarantees imagine of error. Imagine most zero oh .054. Each of the we yes confidence. Mhm Intervals. Each of the three confidence intervals. So on this one will start by saying we have three industries. And so what we're saying P. R1 equals two 0.37 P. R two equals 2.33. And perhaps three is the same as .28. The sample size is 200 At 95% confidence. We know that the ALF of two is the same as mm Z of .25. And this gives us 1.96. This is a critical value critical value from the table. This is a critical body from the table. So start with the government sector and say he had plus minus margin of error for the government. So this is the hot plug plus zero for 2. Um These er for the two is the same but then the proportions are gonna be different. So this is pr 11 minus P. Hard won all over him. Then we'll go ahead and Plug in the numbers so 0.37 plus -1.96 vertical on p hub Is going to be .37 1 -37. All over 200. And so the confidence uh interval in this case for the government becomes estimated as approximately between points. What was 30% to 43%. And then also yeah we have healthcare, we have healthcare and flow healthcare you can see that. Mhm. Won't have pr would rather this should have been He had he had one. Yes that should have been here. And then for healthcare it's we have two plus minus E. Hell and this becomes we had two plus minus Z. L. For two articles Peer to one Rider spear to over end. So pr two is 20.33 plus minus 1.96. Radical points through to me 1 -33 Oliver N. And so the confidence interval okay for health becomes the same as its approximate 0.26 for it to points three line 52. Yeah that in the next part we have education and so P hatch three plus minus E. Education. That's the era of the education level. So this is a three plus minus zero Fortitude Radical Piazza We've won one is pr substitute all over in. So there we have three is 30.28 plus one is 1.96 44. Mhm 0.28 1 -18 all over again. Mhm So the confidence interval and this time it's for education it's going to be approximately the same as .2178 2.342. So that's the first part and then put the Given the heir to the point of five. We understand that mm and is the same as 04 over to want to square that pizza hut one. My spirit. All those E squared and going to go ahead and plug in are specific number assume for estimates some things Period his zero five and so what's going to happen is that Uh huh Gonna take those er for two We want to take this year off of two and say 1.96 squad And then .45 we need another .5. We also have the denominator mhm The denominator when it's when it's rounded up Uh huh You know, just just to make sure we got the right numbers. So we're saying looking for the the margin of error for the three intervals, we're looking for the sample size. So for this one yeah, we already have the margin of error. So this one is also going to be the P had sorry, the p heart The pr is .37 because it's going to be the same for all the industries. So .37 So we change these numbers right here. Change these numbers. So this is Points 37 and this is .37 and so we end up mhm With the denominator 0.0 and All five Squared and so or sample size Sample size becomes approximately descends for 43. So I hope you enjoyed the problem. Well, this one should have been one minus so just remember that this is one mix and this is a square. So I hope you enjoy the problem. Feel free to send any questions or comments. Mhm Hope you enjoy the bargain. Feel free to send any questions or comments and have a wonderful day

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