For the next estimate, derive the formula for the variance of the following estimator, assume that they are three independent samples. (10 pts)
Xi is a random variable with mean = μi and variance = σ2i
P5A2: If three initial samples are taken. With the following
results.
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For the next estimate, derive the formula for the variance of
the following estimator, assume that...
6. Consider the following sample: Xi = -2, X2 = 12. X7-1.5, Xs -0.5, a. Estimate the population mean, μ, using an analogical estimator. b. Estimate the population variance. ơ2, using a biased and an...
6. Consider the following sample: Xi = -2, X2 = 12. X7-1.5, Xs -0.5, a. Estimate the population mean, μ, using an analogical estimator. b. Estimate the population variance. ơ2, using a biased and an unbiased estimator. c. Assuming that the random sample is drawn from a normal population with known variance, σ2-4, construct a 95% confidence interval for the population mean. d. Assuming that the random sample is drawn from a normal population with unknown variance, σ2, construct a...
Estimator properties: 6 Estimators properties 6.1 Exercise 1 In order to estimate the average number of...
Estimator properties:
6 Estimators properties 6.1 Exercise 1 In order to estimate the average number of hours that children spend watching tv, a Bernoulli sample of size n = 5 children was selected from a primary school. Let X be the variable that represents the hours spent watching tv, let E(X)-μ the parameter to estimate and var(X-σ2 the variance. Compare the following two proposed estimators Τι 1. Compare the two estimators for u on the basis of their bias 2....
Question 1: Which of the following would generally cause the variance of the OLS estimator of...
Question 1: Which of the following would generally cause the variance of the OLS estimator of the slope in a regression model to be larger? 1) smaller variance of the error term 2) a larger sample size 3) smaller variance of the independent variable 4) larger variance of Xi ------------------------------------------------------------------------------------------------------------------------------ Question 2: Which of the following is the best description of the sampling distribution of the OLS estimator under the least squares assumptions? 1) it is a Student's t distribution...
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ...
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n We use V, to estimate λ. (a) Show that is an unbiased estimator for λ. (b) Let ơin be the variance of V,, . Show that lin ơi,-
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n...
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2,...
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
x, and S1 are the sample mean and sample variance from a population with mean μ|...
x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate...
Consider a random experiment that has as an outcome the number x. Let the associated variable be X, with true (population) and unknown probability density function fx(x), mean ux. and variance σχ2. A...
Consider a random experiment that has as an outcome the number x. Let the associated variable be X, with true (population) and unknown probability density function fx(x), mean ux. and variance σχ2. Assume that n-2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes xi and x2 Let estimate μ X of true mean #xbe μχ = (x1+x2)/2. Then the random variable associated with estimate μ xis estimator random 1. a. Show the...
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased...
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased estimator for the common standard deviation of two independent sam ples Let Yi, Yi2, ..., Yini denote the random sample of size n from the first population with population mean μ| and population variance σ, and let Y21, Y22, ..., Y2na denote an independent random sample of size n2 from the second population with population mean μ2 and population mean ơ3. Sup- pose that...
Two independent random samples are taken from a normal population with mean 40 and variance 16....
Two independent random samples are taken from a normal population with mean 40 and variance 16. The sample sizes are 5 and 20, and the corresponding sample means are denoted X1 and X2. Determine 1. EX1 - X2) 2. Var (X1 - X2)
1. Consider a random experiment that has as an outcome the number x. Let the associated random variable be X, with true (population) and unknown probability density function fx(x), mean ux, and varia...
1. Consider a random experiment that has as an outcome the number x. Let the associated random variable be X, with true (population) and unknown probability density function fx(x), mean ux, and variance σχ2. Assume that n 2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes x] and x2. Let estimate f x of true mean ux be μΧ-(X1 + x2)/2. Then the random variable associated with estimate Axis estimator Ax- (XI...
6. Consider the following sample: Xi = -2, X2 = 12. X7-1.5, Xs -0.5, a. Estimate the population mean, μ, using an analogical estimator. b. Estimate the population variance. ơ2, using a biased and an unbiased estimator. c. Assuming that the random sample is drawn from a normal population with known variance, σ2-4, construct a 95% confidence interval for the population mean. d. Assuming that the random sample is drawn from a normal population with unknown variance, σ2, construct a...
Estimator properties:
6 Estimators properties 6.1 Exercise 1 In order to estimate the average number of hours that children spend watching tv, a Bernoulli sample of size n = 5 children was selected from a primary school. Let X be the variable that represents the hours spent watching tv, let E(X)-μ the parameter to estimate and var(X-σ2 the variance. Compare the following two proposed estimators Τι 1. Compare the two estimators for u on the basis of their bias 2....
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n We use V, to estimate λ. (a) Show that is an unbiased estimator for λ. (b) Let ơin be the variance of V,, . Show that lin ơi,-
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n...
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate...
Consider a random experiment that has as an outcome the number x. Let the associated variable be X, with true (population) and unknown probability density function fx(x), mean ux. and variance σχ2. Assume that n-2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes xi and x2 Let estimate μ X of true mean #xbe μχ = (x1+x2)/2. Then the random variable associated with estimate μ xis estimator random 1. a. Show the...
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased estimator for the common standard deviation of two independent sam ples Let Yi, Yi2, ..., Yini denote the random sample of size n from the first population with population mean μ| and population variance σ, and let Y21, Y22, ..., Y2na denote an independent random sample of size n2 from the second population with population mean μ2 and population mean ơ3. Sup- pose that...
Two independent random samples are taken from a normal population with mean 40 and variance 16. The sample sizes are 5 and 20, and the corresponding sample means are denoted X1 and X2. Determine 1. EX1 - X2) 2. Var (X1 - X2)
1. Consider a random experiment that has as an outcome the number x. Let the associated random variable be X, with true (population) and unknown probability density function fx(x), mean ux, and variance σχ2. Assume that n 2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes x] and x2. Let estimate f x of true mean ux be μΧ-(X1 + x2)/2. Then the random variable associated with estimate Axis estimator Ax- (XI...
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