(c) Consider two estimators, 01 and 92, of an unknown parameter such that: E(0₁) = 0, E(0₂) < 0 and Var(0₁) Var(0₂). 4 Assuming ₁ and 2 both have normal distributions, sketch on the same graph the sampling distributions of these two estimators, clearly distinguishing between them using labels. You may use ordinary paper no graph paper required.
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- A random sample of n1=17�1=17 securities in Economy A produced mean returns of x̄ 1=5.6% x̄ 1=5.6% with s1=2.3%�1=2.3% while another random sample of n2=20�2=20 securities in Economy B produced mean returns of x̄ 2=4.6% x̄ 2=4.6% with s2=2.3%.�2=2.3%. At α =0.2 α =0.2, can we infer that the returns differ significantly between the two economies? Assume that the samples are independent and randomly selected from normal populations with equal population variances ( σ 12= σ 22)( σ 12= σ 22). T-Distribution Table a. Calculate the test statistic. t=�= Round to three decimal places if necessary b. Determine the critical value(s) for the hypothesis test. + Round to three decimal places if necessary c. Conclude whether to reject the null hypothesis or not based on the test statistic. Reject Fail to Reject9.1) Suppose X1, X2, and X3, denotes a random sample from the exponential distribution with density function shown in the image. a) Which of the above estimators are unbiased for θ? b) Among the unbiased estimators of θ, which has the smallest variance?1. Suppose X1,X2,...,Xn is a random sample of size n > 5 that comes from a distribution with E[X] = μand V ar(X) = σ2. Consider the following two estimators where ̄X is the sample mean:ˆμ1 = (n −3/n) ̄Xˆμ2 = 0.95 ̄X(a) Which of the estimators is an unbiased estimator for μ?(b) Which is the variance of the two estimators?(c) Given the properties above, which one would you prefer? Is there another property that wouldinfluence your decision? Explain.
- a hypothesis test produces a t statistic of t=2.3. if the researcher is using a two tailed test with a=0.05 how large does the sample have to bw in order to reject the null hypothesis?A random sample of n1 = 14 winter days in Denver gave a sample mean pollution index x1 = 43. Previous studies show that σ1 = 19. For Englewood (a suburb of Denver), a random sample of n2 = 12 winter days gave a sample mean pollution index of x2 = 37. Previous studies show that σ2 = 13. Assume the pollution index is normally distributed in both Englewood and Denver. (a) State the null and alternate hypotheses. H0: μ1 = μ2; H1: μ1 > μ2 H0: μ1 < μ2; H1: μ1 = μ2 H0: μ1 = μ2; H1: μ1 < μ2 H0: μ1 = μ2; H1: μ1 ≠ μ2 (b) What sampling distribution will you use? What assumptions are you making? The Student's t. We assume that both population distributions are approximately normal with known standard deviations. The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. The standard normal. We assume that both population distributions are approximately normal with known standard deviations. The Student's t. We…If X1 and X2 constitute a random sample of size n = 2from an exponential population, find the efficiency of 2Y1relative to X, where Y1 is the first order statistic and 2Y1and X are both unbiased estimators of the parameter
- 1) For a two-tailed hypothesis using a z-distribution, find the critical values (z-scores)that will give you a critical region with an alpha of the following values: a) alpha = 0.20 b) alpha = 0.10 c) alpha = 0.05 d) alpha = 0.01 e) alpha = 0.001Mr. Lange draws two independent random samples from normally-distributed popula- tions with unknown population means and unknown population vari- ances. The first sample is found to have x = 78, s21 = 25, and n = 10; while the second sample is found to have y = 74, s2 = 9, and n = 16. Mr. Lange wishes to use Welch’s T Test to determine whether there is enough evidence to conclude the population means differ. Answer the following questions. (a) State the competing theories of this test. (b) Are the requirements for Welch’s T Test met? If not, explain why. (c) Compute the T statistic for Welch’s T Test. Round your answer to three decimal places. (d) What distribution does the T statistic have? (e) Compute α0 as discussed in class. What conclusion can you draw?