Let Y, represent the ith normal population with unknown mean , and unknown variance of for i=1,2. Consider independent random samples, Ya, Y₁2., Yin, of size n₁, from the ith population with sample mean Y, and sample variance S7=E-1(Yu-Y.)². (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of=o=o². (i) Discuss how the distribution of Y₁-₂ can be used to test the equality of the two population means, #₁ and 42, when o=o=o² is known. (j) Define appropriate rejection regions, in terms of Y₁-Y2, for testing Ho: #₁ = 2 against a two-sided alternative hypothesis at the a level of significance.

Let Y, represent the ith normal population with unknown mean , and unknown variance of for i=1,2. Consider independent random samples, Ya, Y₁2., Yin, of size n₁, from the ith population with sample mean Y, and sample variance S7=E-1(Yu-Y.)². (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of=o=o². (i) Discuss how the distribution of Y₁-₂ can be used to test the equality of the two population means, #₁ and 42, when o=o=o² is known. (j) Define appropriate rejection regions, in terms of Y₁-Y2, for testing Ho: #₁ = 2 against a two-sided alternative hypothesis at the a level of significance.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

If X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.
Let Y, represent the ith normal population with unknown mean #, and unknown variance of for i=1,2. Consider independent random samples, Ya, Ya, Yin, of size n,, from the ith population with sample mean Y, and sample variance S² = (₁-P) ². (d) Define V₁, a function of S1, that has a chi-square distribution. (e) Is V₁ a pivotal function to find a confidence interval for o?? Explain with argument. (f) Find a (1-a) x 100% confidence interval for of
Q4 Three electron emitters produce electron beams with changing kinetic energies that are uniformly distributed in the ranges [3,7+45], [2,5+45], and [4,10+45] (Roundup after multiplying by f). Let Y denote the total kinetic energy produced by these electron emitters. (a) Suppose that the three beam energies are independent. Determine the mean and variance of Y . (b) Suppose that the covariance between any two beam energies is -0.5. Determine the mean and variance of Y . (c) Compare and comment on the results in parts (a) and (b).
The quality characteristic of a product Y consists of three components X1, X2, and X3. The distributions of these components are normal with means 100, 2, 1 and variances 1, 1, 1 respectively. The specifications of the product are 90 ± 8. If Y= X1 + -7X2 + -9X3. Products that are below the lower specification limit are scrapped. Find the proportion of scrapped products = Ф(z).
Let S1^2 be the sampling variance for a random sample of twelve values (amount of mercury in the blood) and let S2^2 be the sampling variance for a random sample of ten values (amount of lead in the blood); samples from the same population were used. The population variance for mercury measurements is assumed to be twice the corresponding population variance for lead measurements. We will further assume that S1^2 is independent of S2^2. 1. Find a number b such that P [(S1^2/S2^2)<=b]=0.95 enter such a b to three decimal places b= 2. Consider the number b calculated above, find a number a such that P[a<=(S1^2/S2^2)<=b]=0.90 enter said a to three decimal places. a=
A random sample of 16 adult male wolves from the Canadian Northwest Territories gave an average weight x₁ = 97 lb with estimated sample standard deviation s₁ = 6.7 lb. Another sample of 26 adult male 14 wolves from Alaska gave an average weight x2 = 88 lb with estimated sample standard deviation $₂ = 7.5 lb. USE SALT (a) Let u represent the population mean weight of adult male wolves from the Northwest Territories, and let μ2 represent the population mean weight of adult male wolves from Alaska. Find a 75% confidence interval for μ1-2. (Round your answers to one decimal place.) lower limit upper limit (b) Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 75% level of confidence, does it appear that the average weight of adult male wolves from the Northwest Territories is greater than that of the Alaska wolves? Because the interval contains only…