Let X₁, X2, ..., X₂0 be a random sample from a population X with density 1 20 f(x; 0) = {(1, otherwise where 0 < 0 ≤ is a parameter. The hypothesis Ho: 0 = to be tested against x = 0,1. H₁: p?
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- For the following table of data. x 1 2 3 4 5 6 7 8 9 10 y 0 0.5 1 2 2.5 3 3 4 4.5 5 a. draw a scatterplot. b. calculate the correlation coefficient. c. calculate the least squares line and graph it on the scatterplot. d. predict the y value when x is 11.Let x1, x2, ... , X, be a random sample from N(u, o2), and let 0 = (6x1 - 2x2) -(4x3 -3x4) %3D be an estimator of the parameter µ, calculate the bias of this estimator.Let Xi- NcO, ) andn=2 aud T=X,+ Xz Show that Tis a %3D Sulfficiesry Statistic.
- if x1, x2, ..., Xn be a random sample from Bin(4,0). is T = x UMVUE ? Justify your answer.The proportion of students who get an undergraduate degree in statistics in four years was compared for meals and females. Random sample of 200 male statistics majors and 200 female statistics majors were taken. Choose the appropriate parameter(s) for this situation. [Hint: The .parameter(s) to be used in your hypothesis] ule u 2.00 jo None of these .A O P1-P2.B O H.C O Hi- H2.D O P.E O إنهاء المحاولة . . . PREVIOUS ACTIVITY Note P Type here to search Ae Ps 近13. Let Y,, Y2, ..., Yn denote a random sample from a population with pdf f(yl0) = (0 + 1)yº,0 -1 a) Find an estimator for theta by the method of moments. b) Find the maximum likelihood estimator for theta.
- If x, x2 ... Xn is a random sample from N (µ, o²) popula- tion, find súfficient estimator for µ and ơ.B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.Let Yı, Y2,., Yn be a random sample from the following distribution fy) = {02 for 0sy 202 elsewhere With parameter 0>0. Prove that E1 y? is a sufficient statistic for the parameter 0.
- EXER 6.3 Find the covariance and the correlation coefficient between X and Y, if X and Y are jointly discrete random variables, with joint PMF given by: SHOW SOLUTIONS X\Y 0 1 6 0 28 6 1 28 2 0 333333 28 28 28 2120 28 0Suppose that X₁, X₂, Xn and Y₁, Y2, . Yn are independent random samples from populations with means ₁ and ₂ and variances of and o2, respectively. Show that X - Y is a consistent estimator of μ₁ - 2.B: Let X1,X2, .,X, form a random sample from U(0,0). Find minimal sufficient statistics.