BASICS OF STATISTICAL INFERENCE — DU B.Sc (hons) Statistics Previous Year Question Papers

(b) A random sample X1, Xz, X3, X4 and Xy of size 5 is drawn from a normal population with mean u and variance o?. T1, T½ and T3 are the estimatros used to estimate mean u, where Find 2 such that T3 is an unbiased estimator for u. Are Ty and Tz unbiased estimators? State giving reasons, which is the best estimator among T1, T2 and Tz. (8,7) 4. (a) Describe the method of minimum variance. Prove that MLE of the parameter a of a population having density function f(x, a) = 2ž (a -x); 0 < x < a, for a sample of unit size is 2x, x being the sample value. Also, show that the estimate is biased. (b) Let X1, X2..., Xn be a random sample from N(0, 1) distribution. Obtain MVUE of parameter 0. (8,7) (b) Obtain 100(1 - a)% confidence limits (for large samples) for the parameter 2 of the Poisson(2) distribution. (8,7) 6. (a) Explain the following terms: (i) Type I and Type II errors (ii) The best critical region (ill) Power function of a test (iv) Level of significance. (b) Let X1, X2,..., Xn be a random sample of size n from normal N(0, 02) distribution, where o2 is known. Obtain most powerful test for testing Ho: 0 = 0o against H1: 0= 01(< 00). Also, find the power function of the test. (8,7) 7. (a) Let p be the probability that a coin will fall head in a single toss in order to test Ho: p = ¿ against Hy:p = 5. The coin is tossed 7 times and Ho is rejected, if more than 5 heads are obtained. Find the probability of type I error and power of the test. (b) Prove that, if w be uniformly most powerful test of size a for testing Ho: 0 = 0o against H1: 0 € O1, then it is necessarily unbiased. (8,7) 8. Explain the following terms (Any three) (5x3=15) (i) Fisher-Neyman criterion. (ili) Confidence intervals for large samples (iv) Most powerful critical region and uniformly most powerful critical region.