IfX1 andX2 are the means of independent random samples of sizes n1 and n2 from a normal population with the mean μ and the variance σ ^2 , show that the variance of the unbiased estimator ω·Xbar1+(1−ω)·Xbar2 is a minimum whenω=n1/(n1+n2)
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IfX1 andX2 are the means of independent random samples of sizes n1 and n2 from a normal population with the mean μ and the variance σ ^2 , show that the variance of the unbiased estimator
ω·Xbar1+(1−ω)·Xbar2
is a minimum whenω=n1/(n1+n2)
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- Suppose X and Y are independent random variables with the same mean µ but possibly different variances σx2 and σy2. Imagine that X and Y are the averages of independent samples of size n1 and n2, respectively, taken from populations with the same mean µ and the same variance σ2. Apply below to identify the α that minimizes and the minimum value. Be sure to simplify both the new estimator and the minimum variance. Interpret the result. The variance of αX+(1-α)Y, which also has mean µ, is minimized with α=σy2/(σX2+σy2).If V1, V2, ... , Vn1 and W1, W2, ... , Wn2 are inde-pendent random samples of sizes n1 and n2 from normal populations with the means μ1 and μ2 and the common variance σ2, find maximum likelihood estima-tors for μ1,μ2, and σ2.Two different plasma etchers in a semiconductor factory have the same mean etch rate μ. However, machine 1 is newer than machine 2 and consequently has smaller variability in etch rate. We know that the variance of etch rate for machine 1 is σ12, and for machine 2, it is σ2 = σ 2 12 a . Suppose that we have n1 independent observations on etch rate from machine 1 and n2independent observations on etch rate from machine 2.(a) Show that ˆμ = +− α α X X ( ) 1 2 1 is an unbiased estimator of μ for any value of α between zero and one.(b) Find the standard error of the point estimate of μ in part (a).(c) What value of α would minimize the standard error of the point estimate of μ?(d) Suppose that a = 4 and n n 1 2 = 2 . What value of α would you select to minimize the standard error of the point estimate of μ? How “bad” would it be to arbitrarily choose α= . 0 5 in this case?
- Suppose that X¯, Y¯ be the means of 2 samples of sizes n from a normal population with variance σ^2. Determine n so that the probability will be about 0.95 that the two sample means will differ by less than σ.Suppose that X1, ..., Xn are random variables such that the variance of each variable is 1 and the correlation between each pair of different variables is 1/4. Determine Var (X1 + ...+Xn).If X1, X2, ... , Xn constitute a random sample from anormal population with μ = 0, show that ni=1X2inis an unbiased estimator of σ2.
- Assume that the population consists of only these eleven data points for y and z. Construct one sample for y and one another sample for z by using 5 data points which gives the maximum variance.If you still have some time, you can add your reviews about the contribution of Bessel’s Correction to the variance. Why do we use such correction and can the variance of any possible sample be larger than the variance of the population?A random variable X is uniform from 4 to 8. A Gaussian random variable Y has mean of 10. Approximate the variance ofYY if only 2.5% of its elements is a subset of X.Two random samples of sizes 10, 13 are taken from types of plastics and tested for breaking strength. The test result is that the average strength of the first type 175 and the average strength of the second type is 160. It is known that the two samples have variances 12, 16. The company decides not to use the first type unless its strength exceeds that of second type by at most 9. Should they use the first type or not? (Assume a=0.1)
- Suppose X and Y are two random variables with covariance Cov(X, Y) = 3 and Var(X) = 16. Find the correlation coefficient between X and Y.The teachers at a primary school have devised a particular method to teach arithmetic. They want to find out if this method produces dispersion in mean scores. In order to do so, at first, they took a sample of 25 students who were taught that particular method of arithmetic. Then their test scores were taken into consideration, assuming that the test scores of all the students in that school are approximately normally distributed. Their test score produced an average variance of 180 while the population is assumed to produce a test score of 165. Using a 95% confidence level, can you conclude that the variance of all the test scores of that primary school is different from 165?1.Random sample of size 36 were drawn with replacement from a popolation with mean and variance 0.9, and in each case the sample mean was calculated. Find the variance of X. 2.Random sample of size 9 were drawn with replacement from a popolation with mean and variance 4.2, and in each case the sample mean was calculated. Find the variance of X. 3. Random samples of size 10 are drawn with replacement from a population with mean 4 and variance 6. Let X be the sample mean. Find E(X) and Variance(X).
