Question 4. Suppose for i=1,...,n both the mean and variance are unknown. Based on n=100 sample data, we would like to test vs
a) at a type 1 error level , find a sample statistic T and the rejection region R that correctly controls exactly, i.e., find T and R that satisfy (must be exact in distribution not approximate).
b) Compute the asymptotic power of T, i.e., what does converge to as sample size goes to infinity?
Question 5. Following question 4, except that the test now becomes vs
a) does your previous statistic T still control the type 1 error level correctly for the new hypothesis? Namely, is it true that ?
b)Compute the asymptotic power of T in this case
Question 4. Suppose for i=1,...,n both the mean and variance are unknown. Based on n=100 sample...
Let be i.i.d. . Define the sample mean and the sample variance by and . (i) Find the distribution of and for i = 1, ... , n. (ii) Show that and are independent for i = 1, ... , n. (iii) Hence, or otherwise, show that and are independent. 7l N (μ, σ2) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let X1, X2,.......Xn be a random sample of size n from a continuous distribution symmetric about . For testing H0: = 10 vs H1: < 10, consider the statistic T- = Ri+ (1-i), where i =1 if Xi>10 , 0 otherwise; and Ri+ is the rank of (Xi - 10) among |X1 -10|, |X2-10|......|Xn -10|. 1. Find the null mean and variance of T- . 2. Find the exact null distribution of T- for n=5. We were unable to transcribe this imageWe were...
Suppose is a random sample from exponential distribution having unknown mean . We wish to test vs. . Consider the following tests: Test 1: Reject if and only if ; Test 2: Reject if and only if Find the power of each test at . We were unable to transcribe this imageWe were unable to transcribe this imageHo : θ = 4 We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
3. Let ,..., be independent random sample from N(), where is unknown. (i) Find a sufficient statistic of . (ii) Find the MLE of . (iii) Find a pivotal quantity and use it to construct a 100(1–)% confidence interval for . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
.........,, random sampling of the normal distribution of the unit n, and and ile -1 Let the sample mean and sample variance be respectively. a) b) ile -1 it is independent. c) d) What is the proof ?? We were unable to transcribe this imagen Ν(μ, σ2) T We were unable to transcribe this imageWe were unable to transcribe this imageT We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
Let be a sample (size n=1) from the exponential distribution, which has the pdf , where is an unknown parameter. Let's define a statistic as . Is a sufficient statistic for ? We were unable to transcribe this imagef(x: λ) = Xe We were unable to transcribe this imageT(X) = 1122 T(X) We were unable to transcribe this image
Question 1: Alex was studying how far the University students travelled to take the class on campus. Let W be the distance travelled (measured in miles). This time, a sample of size 50 was collected. He calculated the following statistics based on the sample: Alex wanted to run a hypothesis test with the null and alternative hypothesis as below: Note that is the mean distance travelled by University students to come to campus. Please calculate the test statistic for the...
Using the appropriate model, sample size n, and output: Model: Sample: n=8 S=.5561, = 93.1% , adj = 90.3% 1. Report SSE, , and s as shown on the output. Calculate from SSE and other numbers. Report the total variation, unexplained variation, and explained variation as shown on the output. (Round answers to 4 decimal places.) 2. Report and adjusted as shown on the output. Calculate the F statistic. (Round your answer to 3 decimal places.) 3. Find the...
Let be a sample (size n = 1) from the exponential distribution, which has the pdf where is an unknown parameter. Let's define a statistic as . Is a sufficient statistic for ? We were unable to transcribe this imagef(x: λ) = Xe We were unable to transcribe this imageT(X) = 11>2 T(X) We were unable to transcribe this image
(6 pts) A psychologist who needs two groups of students for a learning experiment decides to select a random sample of female college students and male college students. Before the experiment, the psychologist administers an IQ test to both groups to determine if there is a significant difference in their mean IQ scores. The population standard deviation for this IQ test is know to be for both female and male populations. If the sample mean IQ score of the female...