a)
b)
H0: µ1=µ2=-----=µn
H1: At least one pair of samples is significantly different
p-value<0.00001 (the result is significant)
Question 2. Suppose (X.,X) . FXY, for i = 1, , n. We collect sample data for n-100, obtain sz-2 and Sy-1, and would like to test H0 : Var(x)-Var(y) versus HA : Var(z) Var(y). (a) Using the F test, wh...
All that I need to explain how we get 0.44725 in Part (f) (i) Obtain the expected values of X and Y E(X)-0.29; E(Y)-0.27 (ii) Obtain the variances of X and Y 2.2 . 10-4 Var(X) Var(Y) - 9.874 105 (d) Write down the test statistic for testing Ho: p1 p2 versus Ha: p1 p2 Test St = 2 where the "pooled" standard deviation is n+ n 2 with ơf-Var(X) and σ Var(Y) and nı'n, adjusted accordingly. (e) What is...
Suppose for the two exams in this course, we would like to see if there is any significant improvement from exam 1 to exam 2, i.e., testing H0 : µx ≥ µy vs HA : µx < µy for the average exam scores. Suppose we have n = 36 students, and the sample statistics are x¯ = 21, y¯ = 22, sx = sy = 3 and sxy = 4.5. Compute the p-value using paired two-sample test Suppose we use...
#1 part A.) To test H0: μ=100 versus H1: μ≠100, a random sample of size n=20 is obtained from a population that is known to be normally distributed. Complete parts (a) through (d) below. (aa.) If x̅=104.4 and s=9.4, compute the test statistic. t0 = __________ (bb.) If the researcher decides to test this hypothesis at the α=0.01 level of significance, determine the critical value(s). Although technology or a t-distribution table can be used to find the critical value, in...
Suppose that X1, X2, . . . , Xn is an iid sample of N (0, σ2 ) observations, where σ 2 > 0 is unknown. Consider testing H0 : σ 2 = σ 2 0 versus H1 : σ 2 6= σ 2 0 ; where σ 2 0 is known. (a) Derive a size α likelihood ratio test of H0 versus H1. Your rejection region should be written in terms of a sufficient statistic. (b) When the null...
3. Let X,,X,,..., X, be a random sample from a Gamma 40distribution, where 6>0. we wish to test H0 : θ-1 vs. Hi : θ #1. Show that the likelihood ratio test statistic, A , can be written as A(V) where a. What is the distribution of V? what is the null distribution of what will be the rejection region for an α level test? b. 20 d. 3. Let X,,X,,..., X, be a random sample from a Gamma 40distribution,...
Suppose we would like to test H_0: μ x = μ y . Denote the unpaired two-sample normalized test statistic as t 1 , and the ANOVA F test statistic as \\ t 2 . Suppose for given sample data of equal sample size, we get t 1 = 10 . Compute t 2 . Hint: For equal sample size, establish a relationship between unpaired two-sample test and ANOVA F test.
e. Consider the multiple regression model y X 3+E. with E(e)-0 and var (e) ơ21 Assume that ε ~ N(0 σ21), when we test the hypothesis Ho : βί-0 against Ha : βί 0 we use the t statistic with n-k-1 degrees of freedom. When Ho is not true find the expected value and variance of the test onsider the genera -~ 0 gains 0 1S not true find the expected value and variance of the test statistic. e. Consider...
2) If we now set H(x,y.t)-H0(x,y)+n(x,y,t) and assume that we only have small- amplitude motions with we obtain the linearized shallow-water equations Ot on O a) For the special non-rotating case (f -0 ) with constant depth (Ho - const.) show that the speed of gravity waves is c-VgHo Hint: set v-0 and derive a wave equation for the sea level η b) Given a harmonic wave η(x,t)=Asin(k-or) with amplitude A (again for f-0 and Ho= const.), derive the equation...
2. Suppose we observe the pairs (X, Y), i-1, , n and fit the simple linear regression (SLR) model Consider the test H0 : β,-0 vs. Ha : Aメ0. (a) What is the full model? Write the appropriate matrices Y and X. (b) What is the full model SSE? (c) What is the reduced model? Write the appropriate matrix XR. (d) What is the reduced model SSE? (e) Simplify the F statistics of the ANOVA test of Ho B10 vs....
Question 4. Suppose for i=1,...,n both the mean and variance are unknown. Based on n=100 sample data, we would like to test vs a) at a type 1 error level , find a sample statistic T and the rejection region R that correctly controls exactly, i.e., find T and R that satisfy (must be exact in distribution not approximate). b) Compute the asymptotic power of T, i.e., what does converge to as sample size goes to infinity? Question 5. Following...