Limits theorem

A sample of size n is taken from a population with an unknown distribution, with mean // and standard deviation O. Which of the following statements about the sample mean distribution is true? Select one: a. without Using the Central Limit Theorem, X approximately followsX - N H,- 02 n b. Using the Central Limit Theorem, Xexactly followsX - NCH, 02 n C. without Using the Central Limit Theorem, Xexactly followsX - Nu,- 02 n d. Using the Central Limit Theorem, X approximately followsX - Nu,- nExplain the differences in the sampling distributions of x for large and small samples under the following assumptions. Complete parts a and b, a. The variable of interest, x, is normally distributed. Choose the correct answer bokny. Q A. There are no differences, as both large and small samples will have normal sampling distributions. OF. Small samples will have a normal sampling distribution due to the Contral Limit Theorem, but large samples will not have a sampling distribution. O C. Large samples will have a normal sampling distribution due to the Central Limit Theorem, but small samples will not have a normal sampling distribution. OD. There are no differences, as neither large nor small samples will have normal sampling distributions. b. Nothing is known about the distribution of the variable x. Choose the correct arewer below. Q A. Large samples will have a normal sampling distribution due to the Central Limit Theorem, but the sampling distribution of small samples is unknown. O B. Small samples will have a normal sampling distribution due to the Central Limit Theorem, but the sampling distribution of the large samples is unknown. O C. There are no differences, as both large and small samples will have normal sampling distributions. O D. There are no differences, as the sampling distributions for both large and small samples are unknown.5. Let X1, ....; X100 ~ Exp(B) (i.e., the X, have density f(x) = e-x/8/8). Let 100 X = 100 i=1 (a) Use the Central Limit Theorem to provide an approximate distribution for X. (b) Determine P(|X - 8) 3.1)