251x0835 6/25/08 ECO251 QBA1 THIRD EXAMDue June 27, 2008 Name: _____________________ Exam is normed on 50 points, but any points above 50 wrap around.Part I. (15 points – 1.5 each) Do all the following (2 points each unless noted otherwise). Make Diagrams! Show your work! In particular you must briefly explain how you got the answer to the values of z at the bottom of this page.A. z N~ ( , )0 11. 73.070.3 zP2. 23.1F3. 50.223.1 zP4. The second percentile 98.z 5. A symmetrical interval about the mean with 76% probability.251x0835 6/25/08B. )5,2.1(~ Nx1. 73.070.3 xP2. 23.1xP3. P x 123 2 50. .4. The second percentile 98.x. 5. A symmetrical interval about the mean with 76% probability.251x0835 6/25/08Part II. Do all the Following (35+ Points) Show your work! Neatness counts! Answers of ‘zero’ or ‘one’ especially are unacceptable without an explanation. Do not use one distribution to approximate another without justifying the replacement! Points on an individual question are in parentheses and a running total is in brackets.1. (15+ points)20.05.05.25.005.010.00000005.10.05.010.0542301349yxUse the following events: 9X, which is the event 9x , 44 xX, 33 xX, 11 xX, 00 xX, 33 yY, 22 yY, 44 yYand 55 yY. Find the following.a) Are x and y independent? Why? (1)b) Identify the probabilities below by symbols like ,22YXP 22 YXP or 22 YXP and give their values. (4)i) The conditional probability of Y4 given X3.ii) The joint probability of X3 and Y4iii) The probability of X3iv) The probability of Y4 or (not X3).c) Demonstrate your understanding of Bayes’ rule by finding the conditional probability of X3 given Y4 using, with another probability, two probabilities from b). (2)d) Find 04 xyP. (2) [9]e) Find the variance of y. (1)f) Find the covariance of x and y. (2)g) Find the correlation between xand y. (1)h) Find the variance of yx . (1)g) Find the variance of yx 3(2)251x0835 6/25/082. (10+ points) The education department of the State of Confusion is worried about the size of private college endowments in a period of rising costs. It takes a sample of 9 schools and gets the following numbers (in millions of dollars). Assume that this is a sample taken from the Normal distribution. 60 47 235 900 27 3909 1001 20 833a) Find the sample standard deviation s of the endowments. (2)b) Find a 99% confidence interval for the population mean using the mean and the sample standard deviation that you found in a) (2)c) Repeat b) under the assumption that your sample of 9 was taken in a state in which there are only 20 colleges. (2)d) Assume that, instead of the sample standard deviation you found in a) that the population standard deviation is known to be 1000. Forget about there only being 20 colleges in the population. Find a 95% confidence interval for the population mean (1)e) What you are doing is a simple type of hypothesis testing. On the base of the interval that you found in d) can you say that the mean endowment is significantly different from 1500? Why? (1)f) Let’s continue with hypothesis testing. There are actually 3 ways to do a hypothesis test. One is a confidence interval, which you have already done. You have also found a sample mean for the endowment that is quite a bit below 1500. Let’s say that the education department believes that thepopulation of college endowments is Normally distributed with a mean of 1500 and a population standard deviation of 1000. The sample mean will also have a Normal distribution with a mean of 1500. What will be the standard error (the standard deviation of the sample mean)? (1)g) The critical value method (for the sample mean) is to use the information in f) to construct a symmetrical interval around 1500 with a probability of 95%. If your sample mean does not fall in that interval we can conclude that there is less than a 5% chance that the hypothesis of a mean of 1500 is true. Construct the interval and come to a conclusion. (2)h) The p-value is the probability of getting the value of the sample mean that you actually got or something more extreme. If the p-value is below 5% we can reject the hypothesis that the mean is 1500. Use the values of the sample mean that you found in a) and the distribution that you found in f). Find the probability that the sample mean is as low as or lower than the value that you actually got. To make this probability into a p-value, double it. What is your conclusion? (Because you have to compute a z-score to get this, I call this a test-ratio method. (2)251x0835 6/25/083. (20+ points) Identify the distribution that you are using in each problem. Make it very clear what values of n, p, m or other parameters you are using. If I have to guess what part of what table you used, you will bepenalized. Look at the solved problems for Section L, the solution to Grass3 and ‘Great Distributions’(especially the hints on the 3rd page) before you start.a) On a Wednesday morning trucks arrive at a weighing station at an average rate of 6 per hour. (No! these problems do not have the same answer.) i) What is the probability that exactly eight arrive in a one hour period? (1)ii) What is the probability that more than 8 arrive in a one hour period? (1)iii) What is the probability that more than 16 arrive in a two-hour period? (1)iv) What is the probability of more than 64 in an 8-hour day? (1)v) How many trucks must I be able to weigh in an 8-hour day to be 99% sure that I don’t have to turn any away? (Hint: find the 99th percentile of the distribution in iv) (1) [5]b) Let us assume that we believe that 10% or fewer of our population are unemployed. To test this proposition we take a sample and find that 0x are unemployed. We then compute the probability of0x ormore being unemployed in a sample of that size. If that probability is below 5%, we can doubt that the actual (population) proportion is 10% of fewer. i) So, if we take a sample of 10 and the probability of an individual being unemployed is 10% and we find that three are unemployed, what is the probability of 3 or more being unemployed? Wouldwe doubt the 10% proportion? (2)ii) So, what would we conclude if
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