U7. Let U › N(0, 1) and V ~ x²(v). If U and V are independent than has a student√V/vT-distribution with degrees of freedom. Using this result prove that the statistic Twe used in the two-sample T test, has T-distribution under the null hypothesis. If thetwo sample sizes are m and n, what is the degrees of freedom of T?

Given that U~N0,1 and V~χ2ν If U and V are independent, then UVν has a student T-distribution with ν…

Answered: 7. Let U~ N(0, 1) and V~ x²(v). If U…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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1. A study compared the number of tree species in unlogged forest plots to similar plots logged 8 years earlier. Independent random samples of both logged and unlogged plots were analyzed and the number of tree species was recorded. Tables 1 and 2 present the Shapiro-Wilk normality test for both types of forest plots and the Independent Samples t-test results. Use the information presented in these tables to answer the questions. Table 1: Tests of Normality Tests of Normality Kolmogorov-Smirnov Shapiro-Wilk Forest Plot Statistic df Sig. Statistic df Sig. Tree_Species Logged .181 12 .200 .936 12 .444 Unlogged .110 14 .200 .945 14 .480 *. This is a lower bound of the true significance. a. Lilliefors Significance Correction Table 2: Independent Samples Test Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means 95% Confidence Interval of the Difference Mean Std. Error F Sig. t df Sig. (2-tailed) Difference Difference Lower Upper Equal variances…
Calculate the value of the unpooled test statistic t in each of the following situations. In each case, assume the null hypothesis is Ho: H, - H, = 0. (Ro (a) x, = 38, s, = 8, n, = 64; x2 = 36, s2 = 7, n, = 49 t = (b) The difference in sample means is 43, s.e.(x, - x2) = 27.
Say that we have a null hypothesis that X1,..., X16 that are Normally distributed with mean 0 and variance 1. Call the sample mean X. Consider the test where p-value = 20 (4 X) - 24 (4|X) – 1 We will reject this if p-value < a. a) Show that the type I error rate of this test is a. b) Say that X1,..., X16 that are Normally distributed with mean 1 and variance 1. What is the type II error rate of this test when a = 0.05? •..
5. When Y,,Y,..,Y, is a random sample from N(u,0). we learned in class that the Y-u statistic T = has t distribution with n-1 degrees of freedom, where Y and S are the sample mean and sample standard deviation, respectively. Write out the form of T² and show algebraically that T can be written as the ratio of two independent z random variables, each divided by their respective degrees of freedom. What is, therefore, the distribution of T? (5 pts.)
b. Let X1, X2, X3, and X4 represent the weight of shipment packages at a certain shipment facility. Suppose they are independent normal random variables with means H1 = 3.6, 42=0.9, 3= 1.8, 4 = 7.4 pounds and variances o = = = Find P (2X₁ + 1X2 + 3X3 + 1X4 ≤ 17.6). = 1.
b) Suppose that Y₁, Y₂,..., Y₁0 is a random sample from a bernoulli (P) distribution. It is desired that the null hypothesis Ho: P = 0.5 against the alternative hypothesis H₁: P = 0.1 at a level of significance. If Ho is rejected when Σı Y = 1. i) Determine the level of significance. ii) Find the power of the test.
2
3. If x1, X2, X3, ..., Xn is a random sample from a normal 1 distribution N (µ, 1) show that t= E xí is an unbiased estimator of n i = 1 H² + 1.
3.3.24. Let X1, X2 be two independent random variables having gamma distribu- tions with parameters a₁ = 3, B₁ = 3 and a2 = 5, B2 = 1, respectively. (a) Find the mgf of Y = 2X₁ +6X2. (b) What is the distribution of Y?
Let X1, X2, ... be a sequence of random variables with finite mean u and non-zero variance o?. Let S, = X;. Which random variable has meanequal to 0 and variance - equal to 1? Select one: S-np O a. Zn Vno Sn-H O b. Zn Vno O c. Zn Vño O d. Zn Vno
Assume that you have a sample of n,8, with the sample mean X, 42, and a sample standard deviation of S, 4, and you have an independent sample of n₂-15 from another population with a sample mean of X₂=34 and a sample standard deviation of S₂-5. What assumptions about the two populations are necessary in order to perform the pooled-variance t test for the hypothesis H:1P2 against the alternative H₁: pp and make a statistical decision? Choose the correct answer below A. It is necessary to assume that the populations from which you are sampling have independent normal distributions and equal variances. B. It is necessary to assume that the populations from which you are sampling have equal population means and positive standard deviations. OC. It is necessary to assume that the populations from which you are sampling have negative ISTAT fest statistics and unequal sample means. OD. It is necessary to assume that the populations from which you are sampling have unequal variances and equal…
Suppose X, Y, and Z are three independent normal random variables. X has an expected value of 7 and a standard deviation of 2; Y has an expected value of -2 and a standard deviation of 4; Z has an expected value of 5 and a standard deviation of 12. Let T = 3X + 12Y - 2Z Variable X Y Mean(Expected Value) 7 -2 Standard Deviation 2 4 12 Coefficient 3 12 -2 a) What is the expected value of T? b) What is the variance of Z? c) What is the variance of T? d) What is the standard deviation of T? N LO

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