Just B please
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A: Solution: a:
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Just B please
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- Suppose Y₁, Y2,…, Yn is an iid sample from a beta (a, ß) distribution with a = ß = 0, 0 > 0 a. Use the Factorization Theorem to show that n T = []Y;(1 — Y;) i=1 is a sufficient statistic. b. The sample variance n 1 = ΣΥ; - Y)2 n 1 i=1 is an unbiased estimator of the population variance 1 4(20 + 1)* Is S² the MVUE of o2? Explain.2. A new insecticide laboratory test, the immediately after contact. Is this sufficient evidence to support the advertised claim? Use α=0.05. (d) Conclusion for the problem above is is advertised to kill more than 95% roaches upon contact. In a insecticide was applied to 400 roaches and 384 died Reject the null hypothesis and new insecticide does not kill more than 95% of roaches upon contact. Do not reject the null hypothesis and new insecticide kills more than 95% of roaches upon contact. Do not reject the null hypothesis and new insecticide does not kill more than 95% of roaches upon contact. Reject the null hypothesis and new insecticide kills more than 95% of roaches upon contact.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?.
- 6. Find p-value for testing Ho: ß1 = 0 and Ha: B1 # 0. Test hypothesis at a = .1,.05, and .01 7. Find p-value for testing Ho: ß2 = 0 and Ha: B2 # 0. Test hypothesis at a = .1,.05, and .01 %3D3-4.3 Arandom variable X has a variance of 9 and a statistically independent random variable Y has a variance of 25. Their sum is another random variable Z = X + Y. Without assuming that either random variable has zero mean, find a) the correlation coefficient for X and Y b) the correlation coefficient for Y and Z c) the variance of Z.B2. Consider the testing problem where we have observed a random sample x₁,...,n from the model X₁,..., Xn ~ N(µ, o²), where the X, are independent, and we want to test the hypothesis Ho: o² = o against the alternative H₁: 0² o. For the test we use the test statistic n y = Σ i=1 (x₁ - x)² 0² where is the sample mean of the ₁. (a) What are the type I errors and type II errors in this testing problem? (b) If Ho is true, what is the distribution of the random variable Y corresponding to y? (c) Assume that n = 20 and we reject Ho if and only if y> 32.852. What is the probability of a type I error in this case? (d) now that H₁ is true for the test in part (c). Draw a sketch of the type II error as a function of o2. (There is no need to calculate explicit values.) - (e) Explain why the test described in part (c) is not a good test for testing the hypoth- esis Ho: 0² o against the alternative H₁: o² o2. How would you improve the test?
- Suppose E(X) = 1.50, E(Y) = 0.55, E(XY) = 0.80, Var(X) = 0.25, and Var(Y) = 0.2475 for two random variables X and Y. Which of the following can be said about the relationship between the X and Y variables? note: no explanantion needed just the answer A:weak positive relationship b:weak negative relationship d:strong negative relationship c:strong positive relationship2)Consider a random variable X with MGF M(t) = e3t+4t² Find the mean and variance of X.3.1 Solve the below problem Let Y,, Y2, and Y, be random variables, where E (Y,) = 1, E (Y2) = 2, E (Y3) = -1, V(Y,) = 1,V(Y2) = 3, V(Y3) = 5, Cov (Y1, Y2) = -0.4, Cov(Y1, Y3) = 1/2, and Cov(Y2, Y3) = 2. Find the expected value and variance of U = 2Y, – 2Y 2 + 2Y3. (2.5+2.5) 3.2 Complete the below statement A multinomial experiment is a generalization of the experiment.
- 0. X and Y are two random variables with E(X) = 5, E(Y) = 10, o = 4 and o? = 9 respectively. The correlation coefficient between X and Y is 0.5. New variable U and V are defined as U =X+Y and V = X-Y. Find the correlation coefficient between 'U' and 'V'. %3D %3D5. Out of random sample of n them believe in extraterrestrials. A random sample n2 = 100 adult Americans who did not attend college, r, = 37 of 100 adult Americans who did attend college, r, = 47 of them claim to believe in extraterrestrials. Does this indicate that the proportion of people who attend college and who believe in extraterrestrials is higher than the proportion who did not attend college? Use alpha = 0.01. %3D1. (a) Let X {X1, X2, , Xn} be a random sample from a normal distribution .... with mcan u and variance o. i. If u is known and o? is unknown, show that (X, -p)² is a sufficient i=1 statistic for o². Show all steps of your working. ii. If u and o2 are both unknown, show that X (i.e. the sample mean) and S (i.e. the sample variance) are jointly sufficient for u and o?.