= 3. Suppose Y is an i.i.d sequences of random variables, such as Y₂~ N(0, 1) for i 1,..., n. Let us consider the following estimators: 0₁ n- Yi, (a) Determine the mean of the estimators; (b) Determine the variance of the estimators; (c) Are the estimators biased? 1 6₂ = (3₁+ Yn)

7. For the following data: 9, 5, 10, 7, 9, 10, 11, 8, 12, 7, 6, 91. Compute the z-score for the raw score of 10.2. Compute the z-score for x = 6. 8. Find the probabilities for the following z-scores:1. p(z < 0.50)2. p(z > -0.25)3. p(z < -1.35)4. p(z > 1.68) 9. Given a distribution with ? = 225 and ? = 15, find the following probabilities:1. p(X > 227)2. p(X > 212)3. p(X < 230)4. p(X < 220)

. Given a normal random variable X with mean μ = 100 and variance o² = 49 if we randomly observe X=30, is this an unusual observation? (a) This would be an unusual observation because it lies less than 3 standard deviations away from the mean. (b) This would not be an unusual observation because it lies less than 3 standard deviations away from the mean. (c) This would be an unusual observation because it lies more than 3 standard deviations away from the mean. (d) This would not be an unusual observation because it lies more than 3 standard deviations away from the mean.

Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. Consider the following estimator of μ: �"!= 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4.What is the variance of � "!