Let X1, X2, . . . , Xn be a random sample of size n from a normal population with mean µX and variance σ ^2 . Let Y1, Y2, . . . , Ym be a random sample of size m from a normal population with mean µY and variance σ ^2 . Also, assume that these two random samples are independent.
H0 : σX = σY versus H1 : σX > σY
at 100α% level of significance. Find a test statistic and its sampling distribution under H0. What is the rejection region?
H0 : µX = µY versus H1 : µX not equal to µY
at 100α% level of significance. Assuming that σ 2 X and σ 2 Y are known, find a test statistic and its sampling distribution under H0. What is the rejection region?
H0 : µX = µY versus H1 : µX < µY
at 100α% level of significance. Assuming that σ 2 X and σ 2 Y are unknown but equal, find a test statistic and its sampling distribution under H0. What is the rejection region?
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...
Q6: Let X1, ..., Xn be a random sample of size n from an exponential distribution, Xi ~ EXP(1,n). A test of Ho : n = no versus Hain > no is desired, based on X1:n. (a) Find a critical region of size a of the form {X1:n > c}. (b) Derive the power function for the test of (a).
5. (10 points) Let X1,... , Xio be a random sample of size 10 from a Poisson distribution with mean θ. The rejection region for testing Ho :-0.1 vs. 1.1: θ-0.5 is given by Σ"i z > 4. Determine the significance level α and the power of the test at θ : 05. 5. (10 points) Let X1,... , Xio be a random sample of size 10 from a Poisson distribution with mean θ. The rejection region for testing Ho...
Let X1, X2,.......Xn be a random sample of size n from a continuous distribution symmetric about . For testing H0: = 10 vs H1: < 10, consider the statistic T- = Ri+ (1-i), where i =1 if Xi>10 , 0 otherwise; and Ri+ is the rank of (Xi - 10) among |X1 -10|, |X2-10|......|Xn -10|. 1. Find the null mean and variance of T- . 2. Find the exact null distribution of T- for n=5. We were unable to transcribe this imageWe were...
Let X1, . . . , Xn be independent Gamma(2, θ) random variables. The goal is to test H0 : θ = 2 versus H1 : θ not equal to 2. (1) Find the test statistic Λ. (2) Derive the rejection region of the corresponding LRT
#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...
Please answer both and label them clearly Let X1, X2,..., Xn be a random sample from a normal population with mean y unknown and standard deviation o known. 1. At significance level of a, find the rejection region (decision rule) for the following hypotheses. Họ: A = 90, Ha : < 0. 2. For the rejection region (decision rule) in (1) find the B when = Mi (ui)).
A sample of 1000 observations taken from the first population gave x1 = 290. Another sample of 1200 observations taken from the second population gave x2 = 396. a. Find the point estimate of p1 − p2. b. Make a 98% confidence interval for p1 − p2. c. Show the rejection and nonrejection regions on the sampling distribution of pˆ1 − pˆ2 for H0: p1 = p2 versus H1: p1 < p2. Use a significance level of 1%. d. Find...
Let X1, X2, . . . , Xn be IID N(0, σ2 ) variables. Find the rejection region for the likelihood ratio test at level α = 0.1 for testing H0 : σ2 = 1 vs H1 : σ2 = 2.
A random sample of size 15 is obtained from a normal population yielding a sample standard deviation of 20. Test the null hypothesis that the unknown population variance is greater than or equal to 162 versus the alternative that the unknown population variance is less than 162 using a 5% level of significance a. Set up the null and alternative hypotheses, clearly defining any unknown parameters. Note the “=” value is always in the null hypothesis. b. Find a test...