251x0835 6/25/08
ECO251 QBA1
THIRD EXAM
Due 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 (2points 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.
1.
2.
3.
4. The second percentile
5. A symmetrical interval about the mean with 76% probability.
B.
1.
2.
3.
4. The second percentile .
5. A symmetrical interval about the mean with 76% probability.
Part 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)
Use the following events: , which is the event , , , , , , , and .
Find the following.
a) Are x and independent? Why? (1)
b) Identify the probabilities below by symbols like or and give their values. (4)
i) The conditional probability of Y4 given X3.
ii) The joint probability of X3 and Y4
iii) The probability of X3
iv) 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 . (2) [9]
e) Find the variance of . (1)
f) Find the covariance of and . (2)
g) Find the correlation between and . (1)
h) Find the variance of . (1)
g) Find the variance of (2)
2.(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 20 833
a) Find the sample standard deviation 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 the population 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)
3. (20+ points) Identify the distribution that you are using in each problem. Make it very clear what values ofn, p, m or other parameters you are using. If I have to guess what part of what table you used, you will be penalized.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 8arrive 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 are unemployed. We then compute the probability of or more 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? Would we doubt the 10% proportion? (2)
ii) So, what would we conclude if 5 out of a sample of 25 were unemployed? (Iwant calculations – not opinions.) (1)
iii) Out of 100, what is the lowest number that could be unemployed that would make us doubt the 10% population proportion? Why? (2)
iv) Out of a sample of 250, you find that 34 are unemployed. To figure out the probability of 34 or more being unemployed can you use the Poisson distribution? Why? What is the probability? (2)
v) Out of a sample of 250, you find that 34 are unemployed. To figure out the probability of 34 or more being unemployed can you use the Normal distribution? Why? What is the probability? On the basis of iv) and v) can you doubt the hypothesis? (2) [14]
c) Assume that 2% of the cattle in the UShave mad cow disease.
i)A slaughterer takes a random sample of 10 cattle from the entire US herd, tests it and concludes that, since none of the cattle in the sample have mad cow disease, there is no problem. What, in fact, is the chance that, if 2% of the population is infected, at least one out of the sample is infected?(2)
ii) Assume that you are inspecting a large number of cows, what is the chance that the first infected cow will occur in the first 100 cows you inspect? In the first 1000? (2)
iii) Now assume that the slaughterer buys 100 cattle, that 2% of the 100 are infected and that the slaughterer takes a sample of 10. Is the chance that at least one of the cattle is infected higher or lower than in i)? Show your work!!! (2) [20]
iv) (Extra credit) If, in fact 2% of the US herd is infected, how large would a sample have to be for there to be a 95% chance that at least one cow in the sample is infected? (2)
d) The amount of time it takes to inspect a vehicle has a mean of 23 minutes with a standard deviation of 4 minutes. An inspector is scheduled to inspect 100 vehicles in 40 hours (2400 minutes). What is the chance that the inspector will finish the inspections in the allotted time? (3)
e) (Extra credit) According to Ken Black, amanufacturer has determined that in a 20-minute interval 1.38 defects occur in a manufacturing process. This means that the average time between defective items is minutes. An inspector observes the line for 15 minutes. What is the chance that the inspector will observe a defective item? (2)
4. (10+ points)
a) A bank knows that 90% of its borrowers have been employed at their current jobs for 3 years or more. 5% of the borrowers that have been at their current jobs for 3 years or more are behind in their mortgage payments. However, 15% of the borrowers who have not been at their jobs for 3 years or more are behind in their payments. Let B be the event that a borrower is behind in mortgage payments. Let E be the event that the borrower has been at the job for 3 years or more. What proportion of the people who are behind in mortgage payments have been at their jobs for 3 years or more? You should start by identifying the probability that I have requested in terms of B and E. Then identify 90% and 5% the same way. (4)
b) As everyone knows, a jorcillator has two components, a phillinx and a flubberall. It seems that our new jorcillator only works as long as both components work (so that it fails in the first period if either component fails).
The probability of the phillinx failing is given by a continuous uniform distribution with a lower bound of 11 and an upper bound of 31. For example, if the life of the phillinx is represented by , the chance of the phillinx failing in the first ten years (Period 1) is and the probability of it failing in the second ten years (Period 2) is (NOTE). The flubberall also follows the continuous uniform distribution with and .
In order to maintain my sanity, use the following events. Period 1 is the first ten years, period 2 is the second ten years and period three is happily ever after.
Failure of the phillinx in period 1, 2, 3 are events and
Failure of the flubberall in period 1, 2, 3 are events and
Failure of the jorcillator in period 1, 2, 3 are events and.
i) What is the probability that the phillinx will fail in period 1? Period 2? Period 3? (1.5)
ii) What are the mean and the standard deviation of the failure time for the phillinx? (1)
iii) What is the probability that the Flubberall will fail in period 1? Period 2? Period 3? (1.5)
iv) What is the probability that the jorcillator will fail in the first period? (1)
v) What is the probability that the jorcillator will fail in the second period 2? (1)
vi) What is the probability that the jorcillator will fail in the third period? (1)
If you haven’t figured it out already, one of the easiest ways to do this is to make a joint probability table. Put the events across the top. Put the events down the side. Figure out what the probability of the joint events must be if they are independent. Now make a similar table. This time, instead of probabilities, fill in the period in which the jorcillator fails.