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Abor? gaddilg bunvcy of 1037 edults from the US age 65 and over; 643 were concem Find concerued aboue point estimate for the population proportion of those getting...

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Abor? gaddilg bunvcy of 1037 edults from the US age 65 and over; 643 were concem Find concerued aboue point estimate for the population proportion of those getting the flu_Construct 957 confidence interval for the population proportion: What does this interval say you?Find the minimum sample size needed t0 estimate the population proportion at thc 99r confidence level in order t0 cusure that the e timate accurate within of the population proportion_Tbe Chamber of Commerce of Tampa Bay; Florida:

abor? gaddilg bunvcy of 1037 edults from the US age 65 and over; 643 were concem Find concerued aboue point estimate for the population proportion of those getting the flu_ Construct 957 confidence interval for the population proportion: What does this interval say you? Find the minimum sample size needed t0 estimate the population proportion at thc 99r confidence level in order t0 cusure that the e timate accurate within of the population proportion_ Tbe Chamber of Commerce of Tampa Bay; Florida: would like to estimate the average amount of money spent by visitor in Florida with 952 confidence: Assume that the underlying distribution normal and the standard deviation is 5200. What sample size would be necessary for the Chamber of Commerce to meet its objective in estimating this average amount if population mean is within S20 of the sample mean?



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Use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion $p ;(b)$ identify the value of the margin of error $E ;(c)$ construct the confidence interval; (d) write a statement that correctly interprets the confidence interval. The drug Eliquis (apixaban) is used to help prevent blood clots in certain patients. In clinical trials, among 5924 patients treated with Eliquis, 153 developed the adverse reaction of nausea (based on data from Bristol-Myers Squibb Co.). Construct a $99 \%$ confidence interval for the proportion of adverse reactions.

Let us look at this question. They does a three body temperatures in Appendix B includes 106 body temperatures of adults for day two at 12 Onda. They vary from a low of 96.5 to 99.6. So what is it in 99.6 99.6 minus 96 point 96.5 96.5. What does that stand out to be Over here. 99.6 less 96.5. This is 3.1. This is 3.1. This is my range. And what is going to be my Sigma? My signal is going to be this divided by four. I'm estimating this. So what is it this going to be? This is going to be zero point seven 75 This is 0.775 Is this right? Yes, this is right. 0.775 This is going to be my Sigma for the first calculation. What does this say? This says I want the sample mean within 0.1. Okay, so this is 0.1 is equal toe. What is my Alfa 98% confidence level, which means my Alfa by 20.1 this is going to be 0.1 on my critical value is 2.3263 2.3263 multiplied by What is this sigma 0.775 upon route 10 upon And now let's use the calculators. You don't want to answer in five multiplied by 2.3263 divided by 0.1 and this is square. So this is 3 25.3 which means that end is 3 26. This is the value of end. The minimum sample size requires this. The second one is a zoom Sigma is equal to 0.6 to a zoom Sigma is equal to 0.62 So this is going to become 0.62 now 0.6 to 0.62 And in this case, what is going to be my end is 0.62 to 2.3263 divided by 0.1 and squaring This is 20.0 do what I need and is equal to 209 and is equal to 209 Okay, Which result is likely to be better again. The second one is likely to be better, because in this we're using the sample standard deviation in the first one. We are only estimating it by dividing by food, so this second result is likely to be better.

We are very familiar with the methods. By now we want to construct the confidence level of 99 point Sorry off 99% and we have 20 different readings. So in order to control the confidence interval, what I've done is I've again simply input the values over here. These you can see we have 20 values. We are example mean a study 2.5 and a standard deviation is 20.3, you know, confidence level of 99. I select 99 this is the confidence interval that I get 20.8 to 44.2 20.8 20 pointed 20.8 to 44 point to This is my 99% confidence in trouble. Alright. Could also have done this by using the same old formula here and would have become 20 degree of freedom would be 19. Substitute these values Alfa. I would have become 0.50 point 00 fight and put the values in this formula and we come to the same result. This one now it is asked. Does this say? What is the question? There's the conference interval. Give us a good information about the population of all cancers off the same pretty brands that I consumed. No, because we have taken only one can off every brand in this, and only one can cannot be a representative off all the cans off that particular brand. So the answer is no. And we don't even care if this is normally distributed, because this is simply not going to give us a good idea off the, you know, off the mean off the mean off the mean of what we are. Yeah, the mean off caffeine for 12 rounds off a drink. So we don't even care if this is normally distributed.

Are We want to find the sample mean extra the sample standard deviation s and construct the 90 confidence interval about the population of meaning you or data below. Reassuming population is normally distributed. And once we compute these particular values, we want to interpret our findings to start off with, let's find, expiring s. Remember that to do so we use appropriate definitions for X bar that is the some of the data X I divided by the end. The number of data points. This computer expert equal 62.3 Ash is defined as the square root of the sum of deviations about the moon square, divided by n minus one. Which in this point computes to 8.0. Next let's find the TC value or rather the critical T value from the student's T distribution needed to compute this confidence interval to do so, we have to use a tea table, A tea table takes as input to important things, the degree of freedom in this case nine and the confidence level in this case .9 to output the correct value, we find we use a tea table either from google or on a textbook and doing so, we find T C. Equals 1.833 next we want to compute the margin of Arafat's interval, given by the formula on the left, plugging in T C. S and end for this problem. We obtain E equals 4.6 and we can compute the confidence interval. Now, given the following formula experiments, E is less than you as less than expert plus E plugging in our experts and our E gives confidence interval 57.7 is less than you is less than 66.9, which we can interpret to mean. We are 90% confident that the new the population mean is between 57.7 and 66.9.

Once again, welcome to a new problem. This time we're dealing with proportions. And when you think about proportions, you could have a population and the population would have a population proportion population proportion and the population proportion would be something like the proportion of females in college campus, the proportion of females in a college campus and the population proportion is based on capital P. Um as opposed to the sample proportion, which is jihad, so p heart is X over and this is the sample proportion where you take the uh number of subjects of interest divided by the sample size. And for the most part, there's always a likelihood that you want to do an estimation of the population proportion based on the simple proportion and the estimation can have a margin of error and a confidence level. So for example, if you're 95% confidence Uh then it means your error is 5%. Or sometimes we call that alpha is 5%. And from the distribution point of view, what you're saying is that this middle part is .95 improbability. And then you split the tails, you get .025 on the left tail and then .025 on the right tail. And so the formula for confidence interval uh is the same as the hot plus minus margin of error. Where the margin of error is the same as the alpha over to Radical P Hut one -P had all of the end Members the alpha of the two For 95% confidence. From the tables would be 1.96. So that's the Z score. These twosies course, this would be the negative Z score and then this would be the positive Z score. That's why we have plus and minus over there. Uh So coming back to the problem, coming back to the problem, we have a situation where we're thinking about the report. So given, given the 2003 statistical abstract of the U. S. We see that uh the percentage of mm hmm smokers above 18 and what we're doing and this statistical abstract is too Uh huh. A design study I designed study collecting data own smokers and non smokers on smokers and nonsmokers. Mhm. And there's an estimate, there's an estimate of the proportion who smoke uh early there's an estimate of studies. So in part a. Mhm. Are determined the sample size becoming the sample size to estimate the proportion of smokers and the population uh with 95% confidence level And uh .02 error margin of error. So that's what a And then B mm hmm. Using using sample size requirements, whole suggestions if you want to call it that from part a mm. The study finds 5 20 smokers. Mhm. What is the points estimate of proportion of smokers? Mhm. In the population and the population. And then but see uh determined The 95 confidence interval for smokers in the population, 95% confidence interval of smokers uh in the population. So I want to jump right into it. We have the numbers to work with this problem. So we'll start by saying, Well this was actually 30%. So just remember that Start by saying the P hard in part AP heart is 30%, which is the same as .30. The margin of error is point to remember if you have the hard plus minus margin of error at the same 0.0.30 plus minus point or two. Uh and then your confidence level is 95%. And that means that if you're looking at a distribution, this middle part is .95 And then we split the two sides. This other side is .25 and this other side is also .25. And so he is the alpha over two is the same as 1.96. So we go back and say the margin of error is the alpha over too radical P hut one minus p hurt Oliver in, I want to solve for end. So we divide both sides by Uh zero off over to these two cancel out and we can switch it. So we have radical P heart one minus P hard all over in this is E of zero off over to the square both sides of the equation. And we end up having p hard one minus P hut Oliver in um This is going to give us E squared of a Z off over two squad. And then we multiply both sides by N. And also multiply both sides by the reciprocal of E squared ZR for over two on both sides. That way we can cancel out the uh we can cancel out the sample size is on this side. Then you can cancel out these two on this side. So now we have an appropriate formula for the sample size. So we have ZF over to uh, squirt and then we have a jihad one minus P hurt all of E squared. So if you go back to part A, you're going to see uh want to determine the sample size to estimate the proportion of the population, given that confidence level. So given that confidence level. So, uh, assuming, assuming p heart is 0.45. So we're doing the estimates and assuming Prd's uh huh 0.45. What's going to happen is that I'm going to get And being the same as 0/2 of the two squared and this part Becomes .025. And the reason why it's .25 is because if this is .5 And this fight is also .5, You get the product and you get .025. And then you want to divide that by obviously E squared. The next step would be to plug in the specific numbers 1.96 sq Times .025. And then we want to divide that by e squared, which is a point or two squared. And it's going to give us a sample size of approximately uh 2000 and 17. So that's the first part of the problem. In part a uh in the second part will give an X to be 520 and remember the sample size is to 17. So if you go back to the second question is using that sample size that we just got approximately 2017. The study finds 520 smokers. So what's the point estimate of the proportion? So what's the pr come back here? We know that he had is the same as uh X over. N Soapy hot is going to be the same as X being 5 20 And being 2017. So we get the p had to be equivalent to mm hmm, Approximately 0.25 78. Then the last part of the problem, this is Patsy. Uh we want to find the confidence interval and the confidence interval is p hard plus minus margin of error. already have the P had .2578 and then plus minus the margin of error which was mhm zero for two. So the margin of error. Was he off over to radical uh points. We actually want to displace it here. Zf over to body called p hut one minus p hut all over in. So we're gonna place it here Z alpha is 1.96 and then Radical P Hot is the same as 0.2578 And then 1 -1 0.25 78 11 is still going to 578. And then we want to divide that by 2017 and you can see that the confidence Interval and this is at 95%. If you simplify that On the lower side, it's going to give you 0.2387 and then on the Apple side is going to give you 0.2769. Hope you enjoyed the problem. Feel free to send any questions or comments and have a wonderful thing.


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