Statistics 135: Fall 2004Final ExamName:SID#:There are 10 problems and the number of points for each is shown in paren-theses. There is a normal table at the end. Show your work.1. The designer of a sample survey stratifies a population into two strata,H and L. H contains 100,000 people and L contains 500,000. He decidesto allocate 100 samples to stratum H and 200 to stratum L, taking asimple random sample in each stratum.(a) (5) How should he estimate the population mean?(b) (5) Suppose that the population standard deviation in stratumH is 20 and the standard deviation in stratum L is 10. What w illbe the standard error of his estimate?(c) (5) Would it be better to use proportional allocation?12. George spins a coin three times and observes no heads. He then givesthe coin to Hilary. She spins it until the first head occurs, and endsup spinning it four times in all. Let θ denote the probability the coincomes up heads.(a) (5) What is the likelihood of θ?(b) (5) What is the MLE of θ?23. Suppose that a random variable X has the probability density f (x) =(θ + 1)xθ, 0 ≤ x ≤ 1. Suppose that X1, X2, . . . , Xnare iid with thisdensity.(a) (5) What is the maximum likelihood estimate of θ ?(b) (5) What is the approximate variance of the MLE?34. The intensity of light reflected by an object is measured. Suppose thereare two types of possible objects, A and B. If the object is of type A,the measurement is normally distributed with mean 100 and standarddeviation 25; if it is of type B, the measurement is normally distributedwith mean 125 and standard deviation 25. A single measurement istaken with the value X = 120.(a) (5) What is the likelihood ratio?(b) (5) If the prior probabilities of A and B are equal (1/2 each),what is the posterior probability that the item is of type B?(c) (5) Suppose that a decision rule has been formulated that declaresthe object to be of type B if X > 125. What is the s ignificancelevel associated with this rule?(d) (5) What is the p ower of this test?(e) (5) What is the p-value when X = 120?45. 50 rats were randomly divided into two groups of 25 each. The rats inone group were given steroids. They were then timed running a maze.The rats in the non-steroid group had an average time of 10 secondswith an SD equal to 2 seconds and the rats in the steroid group hadan average time of 9 seconds with an SD equal to 5 seconds.(a) (5) What is the SE of the difference of the two averages?(b) (5) Use the information given to calculate a test statistic for test-ing the null hypothesis that steroids made no difference.(c) (5) What is the P-value of the test?56. (15) Three objects are located on a line at points p1< p2< p3. Theselocations are not precisely known. A surveyor makes the followingmeasurements:(a) He stands at the origin and measures the three distances fromthere to p1, p2and p3. Let these measurements be denoted byY1, Y2, Y3.(b) He goes to p1and measures the distances from there to p2andp3. Let these measurements be denoted by Y4, Y5.(c) He goes to p2and measures the distance from there to p3. Denotethis measurement by Y6.He thus makes six measurements in all, and they are all subject toerror. In order to estimate the values p1, p2, p3, he decides to combineall the measurements by the method of least squares. Using matrixnotation, explain clearly how the least squares estimates would becalculated (you don’t have to do the actual calculations).67. Consider the standard linear model Y = Xβ +, where Y is a n-vectorand β is a p-vector. As usual assume that the errors are uncorrelatedwith constant variance σ2. The least squares estimate of β isˆβ =(XTX)−1XTY.(a) (5) What are the fitted valuesˆY ? Derive their covariance matrix.(b) (5) Show thatPni=1V ar(ˆYi) = pσ2.78. (5) Let Z be a random vector with 4 components and covariance matrixσ2I. Let U = Z1+ Z2+ Z3+ Z4and V = (Z1+ Z2) − (Z3+ Z4). Usematrix methods to find Cov(U, V ).89. (5) Do women have different patterns of work behavior than men?A random sample of graduates of a business school was polled andclassified by gender and workaholism type, resulting in the followingtable:Work Type Female MaleWork Enthusiasts 20 41Workaholics 32 37Unengaged Workers 43 52Relaxed Workers 24 27Disenchanted Workers 37 30How would you test whether there was a gender difference? You do nothave to calculate the value of the test statistic, but explain clearly andunambiguously what the alternative and null hypotheses are, how thetest statistic would be calculated, and how you would assess statisticalsignificance.910. (5) For each of 50 subjects an investigator measures IQ on a standardtest and reaction time to a physical stimulus. Denote the measure-ments from subject i as xi, yi, for i = 1, 2, . . . n. She then calculatesthe correlation coefficient,r =Pi(xi− ¯x)(yi− ¯y)pPi(xi− ¯x)2Pi(yi− ¯y)2Explain clearly and unambiguously how the standard error of r couldbe approximately found by a boots trap