HintlNow, use your previously-computed value as an approximation for sigma, and compute how many ears of the experimental corn the researcher needs in the study. Do...

Construct the indicated confidence interval for the population mean $\mu .$ If convenient, use technology to construct the confidence interval. $$c=0.80, \bar{x}=20.6, \sigma=4.7, n=100$$

Do it or this problem. We need to kind of sample size. They did you estimate so? We know sick, Ma equals. We need to estimate within one unit with a one unit maze, the maximum one. And we have a 98% book only to find the second size using the formula Board lacks being at us. Affected estimates. So this is 98 1 minus 19 eighties. The locals are to half of the one. And then will that do the calculations? He has to be one See for their points. You know, one that we can there in the Capri checking box, norm. And there jazz The value of two point Satti three d sig minus sitting in or work. They have a lover calculated so we can divide both sides by two point. The 1/2 point because to the school and equals three times to sew and requires that you sprinter. Okay, square. There is art. We have 40. Hey, more in 86. You always have wrong back. So you say you need this

Mm. We want to show that for a fixed value of alpha quadrupling and the sample size have the margin of error. E. So for a fixed confidence level, we want to show that quadrupling our sample size has our margin of error. Start off with let's remember a critical definition at the margin of error is equal to your Z. Score, which depends on your alpha level time sigma, your population standard deviation all divided by the square root of em. So now we can consider whether or not Japan have Z. By considering two values of E. We divide by. So for anyone with sample size N and E. To sample size four N. E 2/81 is equal to Z. Sigma over root for in all over the sigma over route. Rz scoring are sigma cancels out and this becomes route and over route foreign. However, we can extract the root for the Namir square to force to so it becomes route and over to root and our roots cancel. And this becomes our desired answer of one half quadrupling and does indeed have E.

Okay, In this problem, you're given a sample. You're given a confidence interval and being asked to find the margin of error and the sample proportion. Now, just to recap, recall that the confidence interval goes from the sample of proportion minus the margin of error to the sample proportion plus the margin of error in a drawing. It might look something like this. You have your confidence interval, and in the very middle you have your sample proportion. And on either side you have your margin of air. Thank you. And in this case, were given the confidence interval 0.512 to 0.596 And finding the population or the sample proportion is quite simple. Soapy. Hot, we can see is that the exact middle between the upper bound and the lower bound of the confidence interval. So he simply find the average between the lower bound and the upper bound, which is 0.512 plus 0.596 divided by two. And this is equal to 0.554 Then, to find the margin of error, we can see that there are two times the margin of error in between the upper and the lower bound and so we can simply find the difference between the upper and the lower bound and divide by two. So this would be 0.596 minus 0.512 divided by two, which is equal to zero point 042 And you can check that this is in fact, the correct answer by reconstructing the confidence interval, using this formula and plugging in these values. So these air your final values for P hat and for the margin of error.

Now let us look at this question. This is a question very similar to the last one. But just these values are going to change. So what is the value of See that I have for this question 0.95 this is 0.95 What is my expert? That is a sample standard SRE sample. I mean, it is going to be 31.39 31.39 My Sigma is 0.80 point eight and my end is 82 is 82. Now what is going to be my Alfa? My Alfa is going to be 0.5 which is nothing but one minus c. So this is 0.5 and my Alfa by two is 0.25 Okay, now again, if I want to calculate this by hand, I can use this formula over here. That is my sample mean plus minus the margin of error. E and I can find my confidence interval which is going to be some lower limit comma a parliament. But if I want to do it by hand, I'm going to use this. Otherwise I will use a calculator. So let us just go to the website that is going to help us find the confidence interval. Our confidence level is 95%. Our X bar is 31.39 X bar is 31.39 later. This is 31.39 Mm. Actually, this is 82. This is our sample size. So this is 82 our sample mean is 31.39 right? 31.39 82. Anna Sigma is 820.8. This is 0.8. All right now I just hit, calculate, and this is my answer. Over here I have 31.21 to 31.56 So this is going to be 31.2168 so that we write. This is to to to 31.562 31.56 This is my confidence interval, which means I'm 95% confidence that might 95% confident that my true population mean is going to lie in this region.