1. Suppose that a random sample consists of three observations, X, X2 and X3, is collected...

 

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1. Suppose that a random sample consists of three observations, X₁, X2 and X3, is collected from a population with mean, and variance o². These observations are independently and identically distributed. To estimate the population mean , consider the following weighted estimator 0: 2 3 4 0 = = 1²/0 X ₁ + 10X²₂² +10² -X3 (a) Find the expected value of 0. Is e an unbiased estimator? Explain. [2 marks] (b) How could you adjust the estimator so that it becomes unbiased? Explain. [2 marks] (c) Find an expression for the variance of the adjusted estimator you suggested in (b). Compare it to the variance of a sample mean with the same sample size of 3. Which estimator is more efficient? Explain your answer. [4 marks] (d) Is the adjusted estimator as good an estimator as the sample mean X? Explain your answer. [2 marks] 1. Suppose that a random sample consists of three observations, X₁, X2 and X3, is collected from a population with mean, and variance o². These observations are independently and identically distributed. To estimate the population mean , consider the following weighted estimator 0: 2 3 4 0 = = 1²/0 X ₁ + 10X²₂² +10² -X3 (a) Find the expected value of 0. Is e an unbiased estimator? Explain. [2 marks] (b) How could you adjust the estimator so that it becomes unbiased? Explain. [2 marks] (c) Find an expression for the variance of the adjusted estimator you suggested in (b). Compare it to the variance of a sample mean with the same sample size of 3. Which estimator is more efficient? Explain your answer. [4 marks] (d) Is the adjusted estimator as good an estimator as the sample mean X? Explain your answer. [2 marks]