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A Phoenix Wealth Management/ Harris Interactive survey of 1500 individuals with net worthof $\$ 1$ million or more provided a variety of statistics on wealthy people (Business Week, September $22,2003$ ). The previous three-year period had been bad for the stock market, which motivated some of the questions asked.$$\begin{array}{l}{\text { a. The survey reported that } 53 \% \text { of the respondents lost } 25 \% \text { or more of their portfolio value }} \\ {\text { over the past three years. Develop a } 95 \% \text { confidence interval for the proportion of }} \\ {\text { wealthy people who lost } 25 \% \text { or more of their portfolio value over the past three years. }}\end{array}$$$$\begin{array}{l}{\text { b. The survey reported that } 31 \% \text { of the respondents feel they have to save more for }} \\ {\text { retirement to make up for what they lost. Develop a } 95 \% \text { confidence interval for the }} \\ {\text { population proportion. }}\end{array}$$$$\begin{array}{l}{\text { c. Five percent of the respondents gave } \$ 25,000 \text { or more to charity over the previous year. }} \\ {\text { Develop a } 95 \% \text { confidence interval for the proportion who gave } \$ 25,000 \text { or more to charity. }}\end{array}$$$$\begin{array}{l}{\text { d. Compare the margin of error for the interval estimates in parts (a), (b), and (c). How }} \\ {\text { is the margin of error related to } \overline{p} \text { ? When the same sample is being used to estimate a }} \\ {\text { variety of proportions, which of the proportions should be used to choose the planning }} \\ {\text { value } p^{*} ? \text { Why do you think } p^{*}=.50 \text { is often used in these cases? }}\end{array}$$

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a. $H_{0} : p \leq 10 \%=0.10, H_{a} : p>0.10$b. 0.13c. There is sufficient evidence to support the claim that Eagle should go national with the promotion.

Intro Stats / AP Statistics

Chapter 8

Interval Estimation

Confidence Intervals

Temple University

Idaho State University

Boston College

Lectures

0:00

03:58

A Phoenix Wealth Managemen…

According to a 2017 Gallup…

01:22

Please provide the followi…

03:12

A recent survey from PaySc…

03:16

okay. In this question, what we have to do is we have to find confidence in troubles. So what is given over here is that there were 1500 individuals with network of dollar one million or more. Okay, so our n a sample size is 1500. Okay, now, let's simply wanted the questions. What's part A. It says the Soviet reported that 53% of the respondents have lost 25% or more of their portfolio in the past three years. So what is P bar fever? This is the proportion, right? Proportion off a sample. In this case, it is 53%. Or I can write this 0.53 These many. This is the proportion off the people who have lost 25% off more of their portfolio value. Now what we have to do is we have to develop in 95% confidential double, which means our Alfa is equal to 0.5 Okay, what is the formula for finding the confidence interval? The formula for finding a confidence interval is fever plus minus rude over fever into one minus fever. This is a sample proportions upon and where n happens to be our sample size. But this thing is multiplied. This thing is multiplied by a critical value. Z star Now what is the star? Zee Star is nothing but see Alfa by doing this case. Alfa by two z Alfa by what is our Alfa Alfa 0.5? So what will be Alfa by two Alfa by two This is going to be 0.25 for E. So how do we get the value of the Alfa? I do. You can either use a calculator or any other statistical tool, and you will find that the value for Z Alfa by two happens to be 1.96 All right, so let us substitute the values. This is going to be 0.53 plus minus rude over 0.53 multiplied by one minus 0.53 which is nothing but 0.47 upon. And and over here I have 1.96 So let me just use a calculator in order to find this. Okay, so this is route over 0.53 multiplied by 0.47 divided by 1500 because N is 1500 and I multiply this thing by 1.96 So this happens to be 0.25 to so this value over here is 0.25 to So this is plus minus 0.53 Right now, this is going to give me a lower value and and upper value. So 0.53 minus 0.252 which is 0.50 for it. Zero point 5048 comma. This is just a woman. 0.53 plus zero point 0 to 5 to what happens to be 0.5552 0.555 to. So this is our confidence interval 95% confidence interval for a part eight. Okay, this is our first answer. Moving on toe part B. What's part B c In part B. It is said that 31% of the respondents feel that they have to save more for retirement to make up for what they lost. And again, we want to develop a 95% confidence interval for the population proportion since we have 95% are Alfa is the same. Which means our Alfa by two is against same that 0.25 and rz Alfa by two is 1.96 We already know the formula that we wrote above, so we're just going to substitute the values here. This is going to be 0.31 plus minus rude over 0.31 multiplied by 0.69 upon 1500 and I multiply this thing by 1.96 Right now I'm going to use the calculator again. So this is route over 0.31 multiplied by 0.69 divided by 1500 multiple 11.96 So this time I have 0.234 So 0.234 This over here is plus minus 0.31 So what I do is 0.31 minus 0.234 with a 0.28660 point 2866 comma. This is 0.3 even, plus 0.2340 point +3334 0.3334 So this is where 95% confidence interval in this case s O. This is the 195% confidence in trouble for the population proportion. Okay. They're not even percent of the respondents feel that they have to save more for the retirement to make up for what they have lost. All right, so this is our answer to part. Be moving on to part, see what is but see. In this case, 5% of the respondents have given $25,000 or more to charity. So again, we want to develop a 95% confidence and double. Okay, so this time R P bar 0.50 point 05 plus minus root over. We have 0.5 multiplied by 0.95 upon +1500 upon 1500 And this is again were deployed by 1.96 right, 1.96 Whatever we have here in plus minus, this is known as the margin of error. So this is Rudo was 0.5 multiplied by 0.95 divided by +500 and I multiply this by 1.96 which happens to be 0.11 So this is 0.11 and I have over here plus minus 0.5 Okay, a soul 0.5 minus 0.11010390 point 039 Coma. What do we have? 0.5 plus 0.110 point 061 Like this is my population proportion. Confidence in trouble. Okay, so this is the 95% confidence in trouble for, you know, the proportion off the respondents who gave $25,000 or more to charity over the previous year. Now, there is also a part D and in part D. They're asking us, how does the margin of error for the different interval estimates is related to Bieber? Okay, So what is the margin of error? The thing that is over here in plus minus. So for the first case, it is 0.252 for part A. It was 0.252 then I think it was 0.234 0.234 And, uh, in the last part, it was 0.10 point 011 If I look carefully in the first part, it was 53% right. RP borrows 53%. Then it came down to 31%. That is point even then 25% 0.5 So I can see that my e My margin of error is decreasing. Right Is this is decreasing with this is decreasing with p bar, okay. And also, another very important thing over here is what they're asking is Why do we choose? Generally, people are as, uh, 0.5, right. This is the question. Why do we choose P bar as 0.5? Well, because it leaves the maximum room for error. That is why I p p. Bar is chosen as zero point five. Okay, so the margin off error is highest. For example, proportions when peak up or pee bar is actually close to 0.5. And I think this completes a problem. Yes. So these would be our ancestors. So I just write This error is maximum error is maximum or I could see that error maximizes error maximizes when r P bar happens to be close to when P bar is close to is close to 0.5 and this is our answer.

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