Homework 9(1).pdf

Homework 9Due on 2018/5/181.(Textbook Section 8.1-1,Page 299)Let X have an exponential distribution with pa-rameter.Suppose that we wish to test the hypothesesH0:1,H1:1.Consider the test procedure that rejects H0if X 1.Compute the size of the test.2.(Textbook Section 8.1-2,Page 299)Suppose that X1,.,Xnform a random samplefrom a uniform distribution on the interval 0,and that the following hypotheses areto be tested:H0:2,H1:2.Let Yn=max(X1,.,Xn),and consider a test procedure such that the critical regioncontains all the outcomes for with Yn 1.5.Determine the size of the test.3.(Textbook Section 8.1-3,Page 299)Suppose that the proportion p of defective itemsin a large population of items is unknown,and that it is desired to test the followinghypotheses:H0:p=0.2,H1:p 6=0.2.Suppose also that a random sample of 20 items is drawn from the population.Let Ydenote the number of defective items in the sample,and consider a test procedure such that the critical region contains all the outcomes for which either Y 7 or Y 1.Determine the size of the test.4.(Textbook Section 8.1-4,Page 300)Suppose that X1,.,Xnform a random sample froma normal distribution for which the mean is unknown and the variance is 1.Supposealso that 0is a certain specified number,and that the following hypotheses are to betested:H0:=0;1H1:6=0.Finally,suppose that the sample size n is 25,and consider a test procedure such thatH0is to be accepted if|Xn 0|1.We shall consider test procedures of the form”reject H0if X c”.For each possiblevalue x of X,find the p-value if X=x is observed.2