1.Suppose we have a random sample X1, . . . , X6 from a uniform(4,θ) distribution with pdf
f(x) = 1/(θ - 4) for 4 < x < θ so that E(X) = (4 + θ)/2.
(a) What is the method of moments estimator of θ?
(b) Is your estimator in part (a) unbiased for θ? Justify your answer.
(c) If the observed data values are
4.3 4.4 4.8 5.2 5.5 6.5
what is your estimate of θ?
2.A statistician has a data set containing n1 = 7 values of a random variable X and n2 = 6values of a random variable Y . She is interested in using the run test to test the equality ofthe distribution functions. That is, she's interested in testing:
H0 : F(z) = G(z)
where F(z) and G(z) are the cdfs for X and Y , respectively.
(a) She decides to reject H0 if the number of runs R ≤ 3. What is the significance level ofher test? In other words, what is the probability of rejecting H0 if it's true?
(b) If the observed samples for X and Y are as follows, what is the statistician's conclusion?
ordered x sample: 3.5 4.3 4.7 5.3 5.4 5.5 6.0
ordered y sample: 4.8 5.1 6.1 6.2 6.5 6.7