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A national health survey suggests that 28% of high schoolstudents in US state they have drank enough alcohol to beconsidered ‘drunk’. The Georgia heal...

Question

A national health survey suggests that 28% of high schoolstudents in US state they have drank enough alcohol to beconsidered ‘drunk’. The Georgia health state department tooka random sample of 1000 high schools students across the state andfound 311 reported they had been ‘drunk’.What is the margin of error if you computed a 99% confidenceinterval instead of a 95% confidence interval?The 95% confidence interval formula for the populationproportion is based on the central limit theorem.

A national health survey suggests that 28% of high school students in US state they have drank enough alcohol to be considered ‘drunk’. The Georgia health state department took a random sample of 1000 high schools students across the state and found 311 reported they had been ‘drunk’. What is the margin of error if you computed a 99% confidence interval instead of a 95% confidence interval? The 95% confidence interval formula for the population proportion is based on the central limit theorem. State the conditions/assumptions to use this formula Interpret, in the context of the question what the 95% confidence intervals means. The drinking age in Denmark is lower than in GA and 85% of students of comparable age as high schools students have reported being “fuld”, which means drunk in Danish. Based on the statewide survey, is it possible that the GA statewide population proportion of high school students who report they have been drunk is same as that in Denmark?



Answers

Use Procedure 12.1 on page 548 to find the required confidence interval. Be sure to check the conditions for using that procedure. A Reader's Digest/Gallup Survey on the drinking habits of Americans estimated the percentage of adults across the country who drink beer, wine, or hard liquor at least occasionally. Of the 1516 adults interviewed, 985 said that they drank. a. Determine a $95 \%$ confidence interval for the proportion, $p$ of all Americans who drink beer, wine, or hard liquor at least occasionally. b. Interpret your result from part (a).

But this problem refers to a survey of 1000 adults, and it was concluded with 95% confidence that from 49% to 55% of Americans believe that big time college sports programs corrupt the process of your higher education institutions. So and was 1000 and we were given a confidence interval at the 95% confidence level of 0.49 up 2.55 So if we think of that in terms of a number line, the low end is 49% the high end is 55% and part A is asking you to calculate the point estimate and the point estimate is identified with the variable p prime. And every time you do a confidence interval, the point estimate is found right in the middle of your confidence interval. So in order to find P prime, we could average the two endpoints of our confidence interval. So we'll take 0.49 plus 0.55 and we'll divide by two. And in doing so, you're going to get a point estimate off 52% or 520.52 The other part of this problem or part A asked you to find the error bound, and the error bound of the proportion is going to be that wiggle room. So it's the distance from the center to each end point of the confidence interval so we can get that by taking the high end of the confidence interval minus the point estimate which gets you three. Or you can go from the point estimate 0.0.52 and subtract the low end. And either way, we get an error bound of 0.3 In part B of this problem, it's asking, can we, with this 95% confidence, conclude that more than half of all American adults believe this? And the answer is no. We cannot conclude that more than half, and the reason being would be because our confidence interval spans from 49% up to 55%. So that means your true proportion can be anywhere in there in that interval. And since the interval goes as low as 0.49 we might be here, which could be less than half so. Therefore, no. We cannot conclude that more than half of all American adults believe this based on this confidence interval as we go into part, C. Parsi is asking you to construct a 75% confidence interval. So we're going to use the P prime that we found in part A and that was 0.52 And we're going to use the fact that we surveyed 1000 people and we need to find the confidence interval well. In order to find the confidence interval, we will need to find the error bound of that proportion, using the Formula Z of Alfa over to multiplied by the square root of P prime times Q prime over N. And in order to calculate the Z score associated with this elf over to, we will need to draw our bell shaped curve, which then puts 75% confidence into center. And our Alfa is the part of the curve that is not accounted for in that confidence interval. So there's 25% of the curve still unaccounted for, and because the bell shaped curve is symmetric, each tale will have half of that or 0.1 to 5. So in each tale we can put a 0.1 to 5 or 12.5%. And then the Z score associated with this left boundary can be found by doing. You're in verse norm on your graphing calculator. And when you use inverse norm, it asks you for three parameters. It asks you for the area in the left tail, which is 1.1 to 5. It asks you for the mean of the standard normal curve. And the standard normal curve has the mean of zero and the standard deviation of the standard normal curve, which is one. So I'm going to bring in my graphing calculator and to access inverse norm, you're gonna hit the second button, the variables button and number three, we're gonna type in the area that's in the left tail, followed by the mean, followed by the standard deviation of the standard normal curve. And we're getting a Z score of approximately negative 1.15 So that negative 1.15 is the Z score associated with the left boundary of the confidence interval. And because of the symmetric nature of the bell, the right boundary is going to be positive 1.15 So, in order to find the error bound of that proportion, we're just going to use 1.15 as RZ and the P prime was 0.52 We're going to multiply that by Q Prime and P Prime and Q Prime must add up to one so Q Prime would be 10.48 and R N was 1000. So our 75% confidence interval error bound is going to be 750.182 So we're not finished yet. We still have to generate our confidence interval. So to generate our confidence interval at the 75% confidence level, we're going to take the point estimate and we're going to subtract the error and we're going to take the point estimate and we're going to add the error. So our estimate was five to so we will subtract 0182 and then we'll take 0.52 and we'll add 0182 So for part C, our confidence interval will be 0.5018 up 2.5382 So the final part of this question is part D. Can we, with 75% confidence, conclude that all American adults believe this, and the answer to that part is yes. We can conclude with 75% confidence that at least half of all Americans believe that big time sports programs corrupt the process off the higher education system. And the reason being would be. Here's our point estimate of 0.5 two, but this time our interval on Lee goes down as low as point 5018 And if the true proportion is found in here, it's always above one half or 10.5. So, yes, we can conclude with 75% confidence that all American adults believe that.

All right, So we're given the annual consumption of alcohol leaders of a sample, 20 European women. And we're gonna make a 95% confidence in our, uh, interval out of that enunciate, please, Andrew. Okay, So who would help if I had a penance? If we run someone variable stats on this subsidizes 20. I mean, is 1 30 the standard deviation. 65.3 nights. All right, so let's find a margin of error. That's t star Times s over the square root of end because we don't really know are population standard deviation. This points us two degrees of freedom of n minus one, which is 19. And this is 95% confidence checking our tea table. We see that, uh, this equal Thio Artie star will be Oh, hello, Artie. Star will be equal to 2.93 Just checking that to make sure. Yes. All right. So, 2.93 I'm 65.39 divided by square root of 20. All right, let's just punch them to a calculator real quick. And when you do, you should get that Your margin of error is equal to 30.6 up. Hello. 30.60 31 All right, so now do you actually construct our interval? All right, So see 1 30 minus 30 points. Some things could be a 99 point 3969 Yes. And adding that there's just gonna be 1 60 0.6031 There's our interval there. Yeah.


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