Hello Coco
I will use the direct (but tedious) method of finding the sampling distribution of the sample means by listing all possible samples, and calculating the mean for each sample. We are taking random samples of size 2 (with replacement) from the set of 4 values, so each sample is an ordered pair of numbers from the set {34,25,19,17}. For example, if the sample is (25,19), then the sample mean is (25 + 19)/2 = 22. The table below lists all results.
Sample Sample Mean
(34,34) 34
(34,25) 29.5
(34,19) 26.5
(34,17) 25.5
(25,34) 29.5
(25,25) 25
(25,19) 22
(25,17) 21
(19,34) 26.5
(19,25) 22
(19,19) 19
(19,17) 18
(17,34) 25.5
(17,25) 21
(17,19) 18
(17,17) 17
Now we will present the sampling distribution of the sample means in a table. We assume that each of the 16 different samples have the same chance of being chosen. To find the probability that the sample mean equals 26.5, for example, we simply have to count how many samples (ordered pairs) have a mean of 26.5. The mean is 26.5 for 2 different samples, (34,19) and (19,34). Therefore
P(Mean = 26.5) = 2/16 = 1/8 = .125
Proceeding in this way, we can complete the sampling distribution of the sample means. The probabilities in the table below are given as both reduced fractions and decimals.
Mean (=X) P(X)=Probability of X
34 1/16 = .0625
29.5 1/8 = .125
26.5 1/8 = .125
25.5 1/8 = .125
25 1/16 = .0625
22 1/8 = .125
21 1/8 = .125
19 1/16 = .0625
18 1/8 = .125
17 1/16 = .0625
Note that the sum of all the probabilities is 1, which is a necessary requirement for a discrete probability distribution. Hope that helps you! Let me know if you have any questions.
William
