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A well-known brokerage firm executive claimed that 60% ofinvestors are currently confident of meeting their investmentgoals. An XYZ Investor Optimism Survey, conduc...

Question

A well-known brokerage firm executive claimed that 60% ofinvestors are currently confident of meeting their investmentgoals. An XYZ Investor Optimism Survey, conducted over a two weekperiod, found that in a sample of 600 people, 61% of them said theyare confident of meeting their goals.Test the claim that the proportion of people who are confident islarger than 60% at the 0.025 significance level.1. Right-tailed, Left-tailed, or Two-tailed? 2. Test Statistic? (round to 3 decimals)3. P-Value? (ro

A well-known brokerage firm executive claimed that 60% of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that in a sample of 600 people, 61% of them said they are confident of meeting their goals. Test the claim that the proportion of people who are confident is larger than 60% at the 0.025 significance level. 1. Right-tailed, Left-tailed, or Two-tailed? 2. Test Statistic? (round to 3 decimals) 3. P-Value? (round to 4 decimals)



Answers

Speaking to a group of analysts in January $2006,$ a brokerage firm executive claimed that at least 70$\%$ of investors are currently confident of meeting their investment objectives. A UBS Investor Optimism Survey, conducted over the period January 2 to January $15,$ found that 67$\%$ of investors were confident of meeting their investment objectives (CNBC, January $20,2006 ) .$
a. Formulate the hypotheses that can be used to test the validity of the brokerage firm executive's claim.
b. Assume the UBS Investor Optimism Survey collected information from 300 investors. What is the $p$ -value for the hypothesis test?
c. At $\alpha=.05$ , should the executive's claim be rejected?

In Question 57. We're told that during an election year we had a sample of 910 adults 503 of which were optimistic about the national outlook. And so, in part, they were asked to give the proportion of the sample that are optimistic. So that is simply P bar is equal to you. 503 over 910 and that comes up to 0.55 to 7 for Part B. We're told that someone wishes to use this poll to indicate that a majority of adults are optimistic about the national outlook, and we're has to devise a hypothesis test test this claim. So let's start with alternative hypothesis. So that is P. So the majority of people is more than 50%. So the alternative hypothesis is that P is greater than 0.5, and the null hypothesis is that P is less than or equal to 0.5, and then in part C, we're as to use the polling data to calculate the P value and then to explain in plain language with the P value means. So our test statistic is said because sample proportions follow a normal distribution. So we have that is equal to this. So we get that said it is equal to 3.18 So our test statistic is 3.18 So if we look that up on the normal standard table, the normal standard table in this and this one only goes up to 3.9 So we know that the area to the left of our test statistic is going to be something greater than 0.999 But this is an upper town test. So the P value is the area to the right of the test statistic. So it means that the P value was something close to zero. So it's a lefty. Value is approximately zero. And then to explain what this means, um, the P value is the probability under the assumption of the null hypothesis that we would get a result at least this extreme. So for this particular example, P values the probability of a sample of size 910 returning a proportion at least as high as 0.55 to 7. Since the probability in this case is extremely small. It leads us to believe that the null hypothesis is not true. And so we reject the null hypothesis in favor of the claim that the majority of adults are optimistic about the national outlook.

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.

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.

Okay, now in sustained care. The first case. Let's take off the sustained care Sustained care. Okay, What is going to be my pick up? My peak at is 0.8280 point 828 Okay, Now my end is 1 19 in is 1 98 and I want a 95% confidence level. Which means my Alfa by two is what my Alfa by two is 0.0 to fight. So Z alphabet to will become 1.96 All right. We already know the formula for e the formula for he is going to be Z Alfa by two multiplied by rude over peak cap into one minus p cap or we also I just ask you cap upon in right. Do we have all the values? Yes. So what is goingto be the value of E? If I substitute these, I get my value of 0.5 to 6. My E in this case turns out to be 0105 to 6. Fine. And how do I write the confidence interval it is given by this. Okay, now these are the two formulas which are required here. Okay, So what is my confidence in double for the first case, 77.54 to 88.6 This becomes 77.54% to 88.6%. Okay, this becomes for the sustained care. Now, what about the standard care for standard here for standard game over here. My P cap is 0.6 to 8. 0.6 to 8. This is my peak app. My in is equal to 1 99. My n is equal to 1 99. And my Z Alfa by two is 1.96 Same as before. Z Alfa by two is equal to 1.96 which is the same as before, right? This is 1.96 So again substituting the values to get the answer for E my He turns out to be 0.672 0.672 What am I using? The same formula. And now I will use this formula to find the interval. And what is the interval that I'm getting in this case? The interval that I get over here is 56.8 to 69.52 56.8% to 69.5 2%. Okay, so what can I say? The confidence interval for the sustained care lies completely above the confidence interval for the standard care, which indicates that the sustained care is more effective than the standard care. Sustained care is more effective, sustained care is more effective and this is going to be my answer.


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