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A. Estimate the value of the population proportion of people who would watch the new series_Population proportionb. Develop a 99% confidence interval for the popula...

Question

A. Estimate the value of the population proportion of people who would watch the new series_Population proportionb. Develop a 99% confidence interval for the population proportion of people who would watch the new series:Confidence intervaland

a. Estimate the value of the population proportion of people who would watch the new series_ Population proportion b. Develop a 99% confidence interval for the population proportion of people who would watch the new series: Confidence interval and



Answers

For the same sample statistics, which level of confidence would produce the widest confidence interval? Explain your reasoning. (a) $90 \%$ (b) $95 \%$ (c) $98 \%$ (d) $99 \%$

All right. And this question we are looking at finding the minimum sample size needed for um achieving a given margin of error. And on this one we're using the same formula as before. So first thing we have to do is figure out our alpha, which if it's a 95% confidence, that's gonna be a 5% alpha. We split that in half to figure out our Z score based off of the appendix table in the back. So I'm gonna go to tea with a tail probability of .025. I'm going to scroll all the way down and I get 1.96 which is always the Z score for a 95% confidence interval, which is the most common one used. So then I'm going to plug in to my formula 1.96 squared times 0.5 Tom's .5 divided by my given margin of error squared. And we can enter that into our calculator and copy and paste will be your friend on these And we get 2401 for part B. Were changing too. A Um 99% confidence in it. Well, it's the only thing changing. So going back to our table or tail probability is half of a percent um which is to .576 2.576 Squared times 0.5 thomas 0.5. Remember we're using our most conservative estimate for P. Hat, which is .5 because it's not given otherwise. And then our margin of error I believe is still oh two Squared. Yes it is. And we can get our sample size clearly goes up rather significantly. We always round up one because we're talking about people here, we round down our margin of error is going to shift down a little bit and we don't want that. And then part C. again a 95% confidence, So 5% alpha, which is the most standard we use, Um which is the 1.96 plugging into our formula and our margin of error is smaller, so it is going to go down and we can copy and paste that into our calculator, like what we had on um part A and just change the point to To be much smaller at .1 and we get 9604 would be are needed sample size for that one.

In exercise six were given a random sample of size 2400, we're going to use these to solve parts A to E and in part A we're supposed to compete the sample proportion Be hard with the characteristic of interest given that 420 out of the 2400 have the characteristic of interest. So for a the hut is equal to 420 Divided by 2400, Which equals 0.1 75. Now, in part B, we're going to verify that that the sample is large enough to use it to construct a confidence interval for the population proportion. Now, for us to verify that the sample is large enough, we shall need to check that uh she heart minus three. Uh Simple standard deviations for the proportion uh coma p hut plus three, simple the sample standard deviation the proportion is within the interval. 0 1. So if we find that this is the case then we can conclude that the the sample is large enough for using the confidence interval for the population proportion. So let's first of all determine the value of Yeah, the sample standard division for the proportion, which is given by the formula squared of PQ over n. And when we substitute, you're going to obtain the following, It's going to be the square root of 0.175 times 0.8- five. All that divided by 2400 and that equals okay, 0.00 77 56. Next you want to get three times the value of this and that equals zero 10- 3- seven. Now we're ready to form that interval in the interval He's going to be given as follows, it's going to be 0.175 0.0- 3- seven. Uh huh. 0.175 plus. 0.023 27. And when you simplify that you've been to obtain yeah 0.1517 And 0.198 three. And you notice that that is a subset of 01, meaning That it is wholly within 01. And therefore we can conclude that the sample is large enough in Patsy, We move on to obtain the 90% confidence interval for the population proportion. Now for this we need to apply the formula Pizza Hut plus or minus The critical value upset multiplied by the square root of PQ over in So let's go on and substitute. So it's going to become 0.175 plus or minus the critical valley for their uh huh For the 0.1 level of significance is going is equal to 1.645. And we multiply by the school to pick you over and which you have already obtained In party and it's 0.0077 56. So this simplifies to 0.175 class of minus mhm zero point zero one 276. And therefore the interval is going to be from 0.16 to two all the way to 0.18 78. Next in part day we shall be calculating ah yeah 99% confidence interval for the population proportion. And so we're going to use the same formula P hut plus or minus. Critical value of that. Well supplied by, pick you over in. Now let's go and do the substitution. So it's equal to 0.175 plus or minus A critical value. For the 99% confidence interval here is 2.5 75. That should be multiplied by the square root of picky over and which was 0.007756. That simplifies to 0.175 plus or minus 0.019 47. And this gives us the interval from 0.1555 two, In part E we're going to comment on why one interval is longer than the other. So when we compare this to intervals, we notice that The interval for the 99% confidence interval is much longer Than the 90% confidence interval. And this leads us to make a conclusion that the greater the confidence needed, the longer the interval just

Another confidence interval. This time we're gonna be looking at 200 people who watch TV in 154 of them. Watch educational TV. Now we want to get a good 90% estimate for the proportion of people in the general population who watch educational TV. So let's start this time by estimating the proportion from what we have to do. This, we're gonna take 154 the number of people who do watch educational TV divided by the total number of people, and we got 77%. That's R P hat estimate, and we can write this as 0.77 if we want. It's gonna be useful later to note that Q hat is one minus p hat. So that's gonna be 0.23 so that these guys at up to 100% great. Now we can get started and find are expected error. The formula for this is our Z term times a square root of P hat times Q hat all over the total number of people end. Since we're working with a 90% confidence interval, RZ value is 1.65 that you can plug in our values for P and Q, that's 0.77 point 23 divided all by 200 the total number of people. And if we plug that in, we get points 05 or 5% to you prefer percentages. Great. Now we can start actually writing our confidence interval. You know that there's a 90% chance that are true. Value of P is greater than P hat plus 900.5 sorry minus 0.5 and that it's less than P hat plus 0.5. No, it's plug in What we have for P hat and we got P is between 0.72 and 0.82 and that was our 90% confidence interval. Again, it could be useful to write this as just P is equal to 0.77 or estimate plus or minus the standard error 0.5 And these are the two ways that you can write your 90% confidence interval

All right, So we're given this set of eight points that comes from our sorry eight data points that come from a norm population. And we have some questions. Ask. So first off, what's your point estimate for the population mean or a point estimate for the population mean sorry. Equal sign is not technically after it. It's gonna be whatever sample mean is, uh, when you have these up, this becomes 10. 18 30 45. 50. There should be an 11 in here. My bad. So 10 18 30 45. 50 uh, 56 67 80. Yep. That was just a sandy check for me. Just making sure I had all the plot point that anyway, So All right. What's your point? Estimates. Come on. There we go. For standard deviation. Well, that's just gonna be the standard deviation of the sample, which, if you do the mathematics for which I'm not going to do by hand right in front of you, you're gonna get three points. 464 All right, now to find hour interval at and 95% confidence. So we want to find out margin of error in party and because we do not have a given population standard deviation. We only have ah, point estimate. Oops. We're going to use a tea interval. So it's gonna be tea time. Standard deviation over skirt and equals eight. We look a t star, we have n equals AIDS. So our degrees of freedom, which is an minus one, is gonna be seven. We check our tea table at seven degrees of freedom for 95% confidence. We get, uh, 2.365 All right, so this equals 2.365 times 3.464 divided by the square root of eight. Punch that in on our calculator and we get So this is about 2.8964 All right, finally, we're gonna constructor confidence Interval. All right, so Oops. There we go. So 10. Plus, he's simple. It's just 12.8 964 and subtracting out. You'll get some 0.1036 There's your interval. My pen's acting weird. Come on. There we go. And there you have it


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