11. Bonus problem (6 points) Assume X,...Xm come from a normal population with unknown variance σ...
Bonus problem (6 points:) Assume X1, ... Xm come from a normal population with unknown variance oị and Y... Y come from another normal population with unknown variance o, and the two samples are independent of each other. Write the 5 steps you will follow to test the hypotheses: Hoi = 1 VS HQI # 1 Make sure to specify the test statistic and the null distribution (don't forget the degrees of freedom), and the rejection region. Hint: Remember that...
DISTRIBUTION OF SAMPLE VARIANCE: Xn ~ N(μ, σ2), where both μ and σ are Problem 4 (25 points). Assume that Xi unknowin 1. Using the exact distribution of the sample variance (Topic 1), find the form of a (1-0) confidence interval for σ2 in terms of quantiles of a chi-square distribution. Note that this interval should not be symmetric about a point estimate of σ2. [10 points] 2. Use the above result to derive a rejection region for a level-o...
Problem 3. Consider two independent samples, X1, . . . , Xm from a N(µ1, σ12 ) distribution and Y1, . . . , Yn from a N(µ2, σ22 ) distribution. Here µ1, µ2, σ12 and σ2 are unknown. Consider testing the null hypothesis that the two population variance are equal, H0 : σ12 = σ22 , against the alternative that these variances are different, H1 : σ12 ≠ σ12 . (a) Derive the LR test statistic Λ
6. Suppose we observe Y,... Yn from a normal distribution with unknown parameters such that Y 24, s2 36, and n 15. (a) Find the rejection region of a level α-0.05 test of H0 : μ-20 vs. H1 : μ * 20. Would this test reject with the given data? (b) Find the rejection region of a level α -0.05 test of Ho : μ < 20 vs. H1 : μ > 20 would this test reject with the given...
. Suppose a random sample of 25 is taken from a population that follows a normal distribution with unknown mean and a known variance of 144. Provide the null and alternative hypotheses necessary to determine if there is evidence that the mean of the population is greater than 100. Using the sample mean, Y, as the test statistic and a rejection region > k}, find the value of k so that α = 0.15. of the form - Using the...
6. Suppose we obeerve Yi,...Yn from a normal distribution with unknown parameters such that Y 24, 236, and (a) Find the rejection region of a level a-0.05 test of Ho : μ-20 vs. Hi :"t 20. Would this test reject with the (b) Find the rejection region of a level α-0.05 test of Ho : μ 20 vs. H: μ > 20, would this test reject with the given data? given data? (c) Will the p-value for the given data...
K=42,n=1,m=18 8. The amount of time it takes a student to solve a homework problem in mathematical statistics (in minutes) follows a normal distribution with unknown mean μ and a variance equal to 9m2. Find the most powerful test to verify the null hypothesis that μ k against the alternative that , 2k on the base of k independent observations, for a significance level of n%. Calculate the power of this test (for the alternative hypothesis). What is the decision,...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we want to test the null hypothesis Ho : Mx = uy against the alternative hypothesis Hi : 4x > uy. Suppose we have i.i.d. pairs (X1,Yı),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Show that exactly) T:= (X-Y)-(ux-uy) - tn-1), Sin (3) where s2 = n-1 [?-,((X; – Y;) –...
24. If the population mean is 0 and the population variance o, 1 (10 points) What is the P (z> 3) a. What is the P (z<2) b. What is the P (-1.5<z <3)? c. What is the P (-2.33cz < 1.25)? d. e. What is the P (-2.33<z and >1.25)? 25. If the population mean is 115 and the population variance σ, 100 (10 points) What is the P (z > 120) a. b. What is the P (2<150)?...
A random sample of size n = 13 obtained from a population that is normally distributed results in a sample mean of 45.2 and sample standard deviation 12.6. An independent sample of size n=17 obtained from a population that is normally distributed results in a sample mean of 51.1 and sample standard deviation 14.9. Does this constitute sufficient evidence to conclude that the population means differ at the a= 0.10 level of significance? Click here to view the standard normal...