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In a survey of 3016 adults aged 57 through 85 years, itwas found that 80.6% of them used at least one prescriptionmedication. Complete parts (a) through (c) below.a...

Question

In a survey of 3016 adults aged 57 through 85 years, itwas found that 80.6% of them used at least one prescriptionmedication. Complete parts (a) through (c) below.a. How many of the 3016 subjects used at least oneprescription medication?b. Construct a 90% confidence interval estimate of thepercentage of adults aged 57 through 85 years who use at least oneprescription medication.c. What do the results tell us about the proportion of collegestudents who use at least one prescription medication?

In a survey of 3016 adults aged 57 through 85 years, it was found that 80.6% of them used at least one prescription medication. Complete parts (a) through (c) below. a. How many of the 3016 subjects used at least one prescription medication? b. Construct a 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who use at least one prescription medication. c. What do the results tell us about the proportion of college students who use at least one prescription medication?



Answers

In a survey of 3005 adults aged 57 through 85 years, it was found that $81.7 \%$ of them used at least one prescription medication (based on data from "Use of Prescription and Over-the-Counter Medications and Dietary Supplements Among Older Adults in the United States," by Qato et al., Journal of the American Medical Association, Vol. 300, No. 24). a. How many of the 3005 subjects used at least one prescription medication? b. Construct a $90 \%$ confidence interval estimate of the percentage of adults aged 57 through 85 years who use at least one prescription medication. c. What do the results tell us about the proportion of college students who use at least one prescription medication?

Now, in this case, it is given that out off. 333,000 and five adults, 81.7%. 81.7% off. People used at least one prescription medication. So what is this number? 81.7% of 3005 off 3005. This is going to be my party dancer to my park. If I use a calculator, this is going to be 81.7. Divided by 100 multiplied by 3005 Or this is approximately 2455 people. This is approximately 2455 people. Yes, this is my answer to Park eight. Now, what is part B five visas? I want to construct a 90% confidence in trouble. Estimate 90% conference and double estimate. 90%. Which means my Alfa by two is 0.5 0.5 This is my Alfa. By do so what is my Z Alfa biting Z Alfa by two is going to become 1.6449 1.6449 We can use a calculator to find this. Okay, Now we have 0.817 This is R P cap 0.817 So what will be a queue? Camp Que cap will be 18.3 or 0.183 Right. This will be 0.183 And we already know the formula for E is the margin of error. He is equal to Z Alfa by two, multiplied by route over. Speak up que capped by N n is 3005, right? And is 3005 adults. Okay, so if I use a calculator to calculate this, this turns out to be rude. Over 0.817 multiplied by 0.183 divided by 3005 And this is multiplied by 1.6449 multiplied by 1.6449 This is zero point 0116 This turns out to be 0.116 Okay, this is my margin off error. Now, if I want to construct a confidence interval what is this going to be first? If this is b the lower limit, what is a lower limit? The lower limit will be 0.817 minus 0.116 This is 0.80540 point 8054 or 80.54% age and 0.817 plus 0.116 182860.8286 This is my confidence in W now. Part C is asking us what the results tell us about the proportion off the college students who use at least one prescription medication. Just a moment. I think over here it is given to us that 3005 adults aged 57 through 85 you know, in a survey off adults, it was found that these many off them used one prescription medication and so on and so forth. So these 3005 subjects are actually adults who are from 57 to 85 years. So what does this actually tell us about the college students? Absolutely nothing, Right? So see is a great question. Question number sees a great question. It tells you absolutely nothing. And this is our answer. This tells us nothing about the college students. Why because these proportions are for adults off 57 to 85 years of age, so this is not answer.

There are often times that we want to find a confidence interval for a population proportion and they'll state what confidence level that we are to use in this problem will take each of the components of forming that confidence interval for a population proportion P. And then also look at what our interpretation is of that confidence interval. So here we have a sample of size and equal 3611 randomly selected. And out of those we have 542 had a particular characteristic. So this could be embedded within a word problem. And then you'd also want to incorporate those specifics of the word problem in with your answer. But here we'll just be doing the numerical calculations. Now. The very first part of the question asks us to find the point estimate for the population proportion and the point estimates notation is P with a hat over it. The notation for the population proportion itself is P without the hat. And for the point estimate for pee it's pee with a hat. Now we find the point estimate from our information from our sample by taking the number in our sample that have the characteristic X. In this case that's 542 and dividing it by the number in our sample. And in this case the number in our sample is 3611. And when you calculate that, do the division you get P hat is about 0.150 part B asks us to verify that we can use the critical value is visible for over two. And our methods that we learned in this section to form the confidence interval for the population proportion and to verify it. There's two things we need to check. The first thing we need to check is if and our sample size times P hat times one minus P hat. multiplying that altogether. Do you get a number that's greater than or equal to time? Well here are n is 3611. Our p hat is 0.150 that we found in the previous part of this example and then our one minus and then the p hat of 0.150 Yeah. For the third factor now multiplying that all together we get the value 460.65 which definitely is a number that's greater than or equal to 10. So yes we've met that condition. The second condition that we need to verify is to check to see if the sample size end if that 3611 for this question is less than or equal to 5% or 0.5 in decimal form times the population um size. And this was particularly tied to um adult american americans 18 years or older. And you can find that with the current census, the a number of adult americans 18 years or over is 200 57,536,000. And when you multiply that out definitely you'll find that 3611 is less than that particular value on the right hand side. So that condition is met as well. So next up let's form the confidence interval. So to find your confidence interval for population proportion, P meeting those conditions, it's P hat minus E to P hat plus E. Where again your P hat is the um X over N. What we found in part A and your E. Is equal to Z sub alpha over two times the square root of p hat times one minus P hat divided by N. Now Z sub alpha over to those critical values. We found them in a previous video how to actually find it for different levels but there's some particular levels of confidence in these examples that occur over and over and over. So it's good to commit to memory that if you're 90% confidence interval is asked with a critical value that's east of alpha over to that value is 1.645 If it's a 95% confidence level when you're critical value is the sub alpha over to. That is 1.96 And if it's a 99% confidence level where your critical value is yours, Isabelle for over two that's 2.575 So try to put those two memory because that will make this process that much faster hours for this particular question is 90% confidence. So he's about to over two is the 1.645 So I have from here our p hat of 0.150 minus. And then the e remember we're going to use our formula for E that I've listed over here to the right so it's the disease of alpha over two for 90%. That's the 1.645 times the square root of He had the 0.15 Okay, times one minus the p hat of 0.15 Yeah. Okay, divided by your sample size end which is 3611. That's the low end. And then to the high end. The upper bound is 0.15 and then plus the 1.645 times the square root of 0.15 times one minus 0.15 mm. And then that divided by and the 3611. And just to color code this the same, I'm gonna make that 0.15 in the parentheses, the same green colors I had with the other ones. Okay, now, when you calculate these through, you'll get a lower bound that 0.14 two and then an upper bound of 0.16 So you just need to run that through your calculator. Be very careful about not putting in too many parentheses, but I'm running it through so that you get the correct values on it. Now, what's our interpretation? We are 90% confident that the population proportion is between 0.14 and 0.16 If you have specifics in the question that you're dealing with, then you actually want to put those words incorporated into describing what the population proportion is focussed on. But here, I wanted to actually show you more about how the calculations are done. Well, I hope this video was helpful and um, look for more videos on others, um, topics, mathematics in general and specifically, and also statistics.

Alrighty. So our population or sample proportion in blue is going to be the X over N. Which is equal to 985 divided by our sample size of 1516. This is a bad equal to 0.65. Make that look a little bit more like a zero. There we go. Beautiful. Rz statistic is going to be uh 5% sipping a thickens level. So then we're looking for 0.025. She tells us that are critical values at 196. Okay. And here we're gonna be finding the upper and lower bounds. So this is going to come out to be approximately An interval of 0626 To a 0.674. So we are 95% confident that the true proportion of all Americans who drink alcohol occasionally is between these numbers right here.

This problem states that 50% of the adult population of the United States takes prescription drugs on a regular basis for adults 65 older. The problem states that 82% take prescription drugs regularly. Wow! For adults 18 to 64 years old, only 49% take prescription drugs regularly. The problem also states the 18 to 64 year olds account for 83.5% of the adult population. Heart A asks what is the probability that a random adult is 65 years old and older? This is equal to the compliment of 18 to 64 year olds and is thus one minus the probability of being 18 to 64 years old. It is therefore one minus 2.835 which is given in the problem and therefore the probability that a random adult, a 65 years old equals 0.165 Part B says that given an adult takes prescription drugs regularly, what is the probability the adult is 65 years old and older? You can calculate this probability using Bayes Theorem the probability of being 65 plus, given that you regularly take prescription drugs is equal to the probability that you regularly take prescription drugs, Given the US 65 years old times the probability of being 65 older. Divide about the probability that you take prescription drugs regularly, given that you're 65 older times the probability of being 65 older, plus the probability that you take prescription drugs regularly. Given your 18 to 64 years old times the probability of being 18 to 64 years old. This is equal 2.82 times, 0.165 divided by pointing to Tom's 0.165 plus 0.49 temps, 0.835 equals 0.2485


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