A simple random sample of size n= 70 is obtained from a population with u = 44 and o = 10. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally distributed? Why? What is the sampling distribution of x? Does the population need to be normally distributed for the sampling distribution of x to be approximately normally distributed? Why? O A. Yes because the Central Limit Theorem states that the sampling variability of nonnormal populations will increase as the sample size increases. O B. No because the Central Limit Theorem states that only if the shape of the underlying population is normal or uniform does the sampling distribution of x become approximately normal as the sample size, n, increases. O C. Yes because the Central Limit Theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x normal, regardless of the sample size, n. O D. No because the Central Limit Theorem states that regardless of the shape of the underlying population, the sampling distribution of x becomes approximately normal as the sample size, n, increases. What is the sampling distribution of x? Select the correct choice below and fill in the answer boxes within your choice. (Type integers or decimals rounded to three decimal places as needed.) O A. The sampling distribution of x follows Student's t-distribution with u- = and o; = O B. The sampling distribution of x is skewed left with p; = and o; = O C. The sampling distribution of x is uniform with p = and o; = O D. The sampling distribution of x is normal or approximately normal with u; = 44 and o; =

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A simple random sample of size n= 32 is obtained from a population that is skewed left with u = 58 and o = 10. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally distributed? Why? What is the sampling distribution of x? Does the population need to be normally distributed for the sampling distribution of x to be approximately normally distributed? Why? O A. Yes. The central limit theorem states that the sampling variability of nonnormal populations will increase as the sample size increases. O B. Yes. The central limit theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x normal, regardless of the sample size,n. O C. No. The central limit theorem states that regardless of the shape of the underlying population, the sampling distribution of x becomes approximately normal as the sample size, n, increases. O D. No. The central limit theorem states that only if the…

3. If x1, X2, X3, ..., Xn is a random sample from a normal 1 distribution N (µ, 1) show that t= E xí is an unbiased estimator of n i = 1 H² + 1.

4. The pmf of a geometric distribution with parameter p is given by S p(1 – p)" n= 0, 1, 2, ... otherwise. Pn = Assume p> 0. Show that the variance is (1 – p)/p². Hint: Σ Eng" (1 – q)² n=0 for |g| < 1.

Only need part f

#4: Suppose that X1,... , X7n is an i.i.d. random sample drawn from a U[a, 1] uniform distribution, where 0a < 1. Suppose that you want to find an unbiased estimator, â, for a. (a) By definition, what relation must be satisfied for â to be an unbiased estimator of a (b) Use the first-order statistic to find an unbiased estimator of a. Recall that the first-order statistic has the p.d.f: п-1 g(yn)nf(yn) f(x)da Уп where f is the p.d.f. of each of the X;'s. (c) Find an alternative unbiased estimator â of a, which is still based on the random sample (hint: think of a common statistic we calculate from the random sample)