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For a large university system, campus life staff are interestedin the proportion of students living on campus who plan to moveoff-campus in the next academic year. ...

Question

For a large university system, campus life staff are interestedin the proportion of students living on campus who plan to moveoff-campus in the next academic year. The staff feels thispercentage has decreased from the consistent 70% it has been inyears past. Current on-campus students are surveyed and 99out of 150 indicate their intention to move off-campus. Atthe 0.05 significance level, use the critical value method to testthe staff’s claim that there has been a decrease.a) Calculate the

For a large university system, campus life staff are interested in the proportion of students living on campus who plan to move off-campus in the next academic year. The staff feels this percentage has decreased from the consistent 70% it has been in years past. Current on-campus students are surveyed and 99 out of 150 indicate their intention to move off-campus. At the 0.05 significance level, use the critical value method to test the staff’s claim that there has been a decrease. a) Calculate the Test statistic b) Find the p-value



Answers

It is claimed that the students at a certain university will score an average of 35 on a given test. Is the claim reasonable if a random sample of test scores from this university yields $33,42,38,37,30,42 ?$ Complete a hypothesis test using $\alpha=0.05 .$ Assume test results are normally distributed. a. Solve using the $p$ -value approach. b. Solve using the classical approach.

Oh we want to answer parts A and B of this question per A. States with no point estimate on the population proportion PP hat what sample size N is needed to be 99% sure that the hat will be within 990.5 from the population proportion P. This question is challenging our understanding of how to derive sample size in order to compute a confidence interval or estimate on P that is within a certain error at a certain confidence. So we need to know the correct formula to solve this problem. That formula is and equals 1/4 of the C over E squared where the sea is a critical. The score associated with confidence is the mark of the Earth. For 0.99 we have point to 0.0.58 and for marginal very specified is 0.5 We have N equals 664. That is plugging and Z of 99 E. We obtain and rounding up 664 as our minimum sample size meeting next party. We compute the same thing, but this time we compute it for known P had equals point by four. That is, we have a point estimate on the population portion of periodicals 40.0.5. For this time we're going to use the formula and equals P hat times one minus P hat Z C over E squared again, we use 0.99 E specified as 0.5 And plugging in. This time we obtain, rounding up and equal 662 as a sample size needed.

Okay, so in this question, they tell us a university has 25,000 students, of whom 17 dozen there under bedroom. It's how the office takes a simple random sample of 500 students, and 357 of them end up being undergraduates. So for party, it says, Um, for the number of undergraduates in the sample, the absurd value is blink. But the expected values blink well. They tell us that, um, there are 350 seven's undergraduates and sample, so it's given the problem and are expected. Value is the percentage of undergraduates, which is 17 1000 over the number of students, which is 25,000. So that's gonna be our percentage of undergraduates and are expected number of undergraduates, though, is that percentage times our number. Our sample size basically so 500. So that ends up just being 68% times 500 um, which is equal to 340. So that's our expected value. Um, of undergraduates in the sample. Her puppy asses for the percentage of undergraduates in a simple, the absurd value is blink but expected value. It's blank. Um, all right. So in terms of percentage. Er well, were given that there are 300 were given that we observe 3 57 You're gonna make us in your color, actually. So we're given that, um we observed 357 undergraduates and were given that their fight were given that there 500 total that are in our sample This end up, um, putting downtick tank leader being 71.4%. That's the percentage of undergraduates that we observed. Um, yeah, but we expected to have this 68% because that's what we just found. So are expected. Are you in terms of percentage is 68% because we have 25,000 students with $17 under bed roots and 17,000 divided by 25,000 is 68%. Um, yeah, I hope this helps

Number 25 Question 18 Given C equal 99% and equal 0.5 formula. Sample size people are known and equal the Alpha over to old power to people queue. People over e squared equal went off over to hold for two people, one minus people over. He's great People are unknown in equal Did all for over two square 0.25 over the square for confidence. Level one minus alpha equals 10.99. That reminds it off over two equals 0.5 using table three A in the appendix Look up 0.5 in the table. The score is then the found that school with the opposite sign or you staple three p So did all for over to equal 2.575 Okay, people is unknown. Then the sample size is round up in equal is it alpha? Over to all power to by 0.25 Overeat square equally 2.575 squared point point 25 over 0.5 square equal 600 64 Question number B Yeah, given C equal 99% equal 0.5 people are equal 15 all percent equal 0.54 formula. Sample size. We will use the above formula. So for confidence, level one minus alpha equals 10.99. That reminds it all for over two equals 0.5 Using table 38 independent. Look at 0.5 in the table. Does it school is then the found that school with opposite signs or use the three p. So is it all over to equally? 2.575 Mhm then in equal, 2.575 squared boy brain four were in 54 multiplied boy, one minus point 54 over 0.5 squared equals 659 mhm.

It is assumed that the house is nearby. The college are, you know, greater in mean value than what is assumed. That is 88950 So this is going to be alternative hypothesis and null hypothesis will be that immune is equal to 89 50 Now, my Alfa for this question is 0.5 I have my sample size of 12. The mean off these 12 is 92460 And my sample center division is 5200 Let's connect E defeatist. So t is equal toe 92460 minus the hypothesized mean, that is 88950 88950 upon s by routine that is 5 to 00 divided by route and I'm going toe. Take the help of the calculator to calculate this root and root and will be Route 12. So this is 92460 minus 88950 Divided by 5 to 00 a bite. Route 12. So this is 1.0 point one. This is T is equal toe zero point 194 This is the value of my T statistic. Now there are two ways that I can solve this. The first is the classical approach of the critical value method. What is the critical value off? T for? Alfa is equal to 0.5 and N is equal to call. First of all, this is going to be a right tail test. So this is a single tail test. So let me just try to find the critical value. I'm going to take the help off an online software in order to do this. Okay, then the time this happens, let's calculate the P value. The P value for T is equal. Do 0.194 and degree of freedom 11. The is equal to 0.194 p is equal to zero point 1 9/4 and B F off 11. B f off 11th at 0.5 And this is of until test The P value is point for two. E can see that my P value my P value in this case the value is equal to 0.42 And since my P values greater than Alfa, my Alfa level is 0.5 I can say that I do not have enough evidence to reject manual hypothesis or I do not have enough evidence to say that the mean is any greater than 8950 So I fail to reject the null hypothesis with this method. What about the critical devalue matter? The critical T value for 0.5 and degree of freedom 11 degree off freedom 11. Sure. Okay, it is 1.796 It is 1.79 60. Critical value is one point 796 And as I can see, that mighty value that I have got is less than 1.796 Again, I fail to reject than a hypothesis in order to show this on a diagram. If this is mighty critical value Mighty star off 1.796 The people that I have is 0.194 which falls somewhere around here. And hence I cannot reject my little hypothesis.


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