Consider the set of odd singledigit integers 1 3 5 7 9 a Mak

Consider the set of odd single-digit integers {1, 3, 5, 7, 9}.

a. Make a list of all samples of size 2 that can be drawn from this set of integers. [Sample with replacement; that is, the first number is drawn, observed, and then replaced (returned to the sample set) before the next drawing.]

b. Construct the sampling distribution of sample means for samples of size 2 selected from this set.

c. Construct the sampling distributions of sample ranges for samples of size 2.

(PLEASE SHOW WORK)

Solution

As we are sampling with replacement, there are 5x5 = 25 possible samples, each with probability 1/25

A. We are asked for the possible samples. They are:

(1, 1), (1, 3), (1, 5), (1, 7), (1,9),

(3, 1), (3, 3), (3, 5), (3, 7), (3,9),

(5, 1), (5, 3), (5, 5), (5, 7), (5,9),

(7, 1), (7, 3), (7, 5), (7, 7), (7,9),

(9, 1), (9, 3), (9, 5), (9, 7), (9,9)

B.

(1, 1) has sample mean (1+1)/2 = 1, so Px-bar(1) = 1/25

(1, 3) and (3,1) have sample mean 2, so Px-bar(2) = 2/25

(1, 5), (3, 3), and (5, 1) have sample mean 3, so Px-bar(3) = 3/25

(1, 7), (3, 5), (5, 3), and (7, 1) have sample mean 4, so Px-bar(4) = 4/25

(1, 9), (3, 7), (5, 5), (7, 3), and (9, 1) have sample mean 5, so Px-bar(5) = 5/25 = 1/5

(3, 9), (5, 7), (7, 5), and (9, 3) have sample mean 6, so Px-bar(6) = 4/25

(5, 9), (7, 7), and (9, 5) have sample mean 7, so Px-bar(7) = 3/25

(7, 9) and (9, 7) have sample mean 8, so Px-bar(8) = 2/25

(9, 9) has sample mean 9, so Px-bar(9) = 1/25

Px-bar(1) = 1/25

Px-bar(2) = 2/25

Px-bar(3) = 3/25

Px-bar(4) = 4/25

Px-bar(5) = 1/5

Px-bar(6) = 4/25

Px-bar(7) = 3/25

Px-bar(8) = 2/25

Px-bar(9) = 1/25

C.

(1, 1), (3, 3), (5, 5), (7, 7) and (9, 9) have range 0, so Px-range(0) = 5/25 = 1/5

(1, 3), (3, 5), (5, 7), (7, 9), (3, 1), (5, 3), (7, 5), and (9, 7) have range 2, so Px-range(2) = 8/25

(1, 5), (3, 7), (5, 9), (5, 1), (7, 3), and (9, 5) have range 4, so Px-range(4) = 6/25

(1, 7), (3, 9), (7, 1), and (9, 3) have range 6, so Px-range(6) = 4/25

(1, 9) and (9, 1) have range 8, so Px-range(8) = 2/25

Px-range(0) = 1/5

Px-range(2) = 8/25

Px-range(4) = 6/25

Px-range(6) = 4/25

Px-range(8) = 2/25