8.3.16. Let (X₁,Y₁), (X2, Y2),..., (Xn, Yn) be a random sample from a bivariatenormal distribution with #1, #2, 0 = 0 = 0², p=, where 1, 2, and o2 > 0 areunknown real numbers. Find the likelihood ratio A for testing Ho: 1 = 2 = 0,0²unknown against all alternatives. The likelihood ratio A is a function of whatstatistic that has a well-known distribution?

Given: Let X1, Y1, X2, Y2,......,Xn, Yn be a random sample from a bivariate normal distribution with…

Answered: 28.3.16. Let (X₁, Y₁), (X2, Y2),...,…

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...

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c) Answer both of the following: i) Consider a random variable X which is normally distributed with mean zero and unit variance. Using the proper statistical table, find the number z such that P(z < X < 2.64) = 0.4760. Give your answer rounded to 2 decimal places. follows at distribution with (2n + 1) degrees of freedom, i.e. Y ~ t2n+1- ii) A random variable Make use of the appropriate statistical table to get the value of n knowing that P(Y 2 3.1824) = 0.025.
Suppose X₁,, Xn from a normal distribution N(μ, o²) where both μ and o are unknown. We wish to test the hypotheses Ho: o² = o vs. Ha: 0² # of at the level a. Show that the likelihood ratio test is equivalent to the x² test.
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…
2.2. Let Y₁, Y₂Yn denote a random sample from a gamma distribution with each Y-gamma (0; B) with known. Find a sufficient statistic for 8.
(e) Show that the sample mean of a simple random sample (drawn without replacement from a normal population) is dis- tributed normally with mean u and variance -² (1--=-1). n where N and n are population and sample sizes respectively. 4 In the 1 1 TI
1.
26 The table to the right shows the cost per ounce (in dollars) for a random sample of toothpastes exhibiting very good stain removal, good stain removal, and fair stain removal. At α=0.01, can you conclude that the mean costs per ounce are different? Perform a one-way ANOVA test by completing parts a through d. Assume that each sample is drawn from a normal population, that the samples are independent of each other, and that the populations have the same variances. Very good stain removal Good stain removal Fair stain removal 0.37 0.75 0.60 0.49 2.66 1.18 0.33 0.46 0.46 1.64 0.33 0.50 0.58 0.41 1.39 (b) Identify the degrees of freedom for the numerator and for the denominator, determine the critical value, and determine the rejection region. The degrees of freedom for the numerator, d.f.N, is ____ and the degrees of freedom for the denominator, d.f.D, is _____ The critical…
16. Let X, X2 X, be a random sample from a normal distribution, X, N(6, 25), and denote by X and S2 the sample mean and sample variance. Use tables from Appendix C to find each of the following: (a) P[3 < X < T. (b) P[L860< 3(X-6)/S]. (c) PIS <319375]
If continuous random variable normally distributed with mu = 256.43 and J = 27.11 , what is P(225.75 <= x <= 300.35) ?
A study of parental empathy for sensitivity cues and baby temperament (higher scores mean more empathy) was performed. Let x, be a random variable that represents the score of a mother on an empathy test (as regards her baby). Let x, be the empathy = 68.36. Another random sample of 31 fathers gave score of a father. A random sample of 37 mothers gave a sample mean of x, X, = 60.06. Assume that o, = 11.55 and o, = 11.45. (a) Let u, be the population mean of x, and let u, be the population mean of x,. Find a 99% confidence interval for u, - Hz. (Round your answers to two decimal places.) lower limit upper limit (b) Examine the confidence interval and explain what it means in the context of this problem. Does the confidence interval contain all positive, all negative, or both positive and negative numbers? What does this tell you about the relationship between average empathy scores for mothers compared with those for fathers at the 99% confidence level? Because the interval contains only…
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?

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