Question

Consider a toy population that consists of three numbers 1,2, and 3.

A. Let X be the randomly selected number (P(X = x) = 1/3) for x = 1,2,3. Find E(X) and V(X).

B. What are the values of the population mean µ and population variance σ^2?

C. Take a random sample of size n = 2 (with replacement). Determine the sampling distribution of the sample mean x̄, i.e. find the probability for each possible value of x̄.

D. Find the standard error of x̄. An observed sample consists of 1 and 2. Based on this piece of information, what is the (estimated) standard error?

Answer 1

Answer:

Given that:

Consider a toy population that consists of three numbers 1,2, and 3.

population unit = 1,2,3 Xi

population size N= 3

population mean  

population variance

A) Let X be the randomly selected number (P(X = x) = 1/3) for x = 1,2,3. Find E(X) and V(X).

P(X = x) = 1/3 , x = 1,2,3

B) What are the values of the population mean µ and population variance σ^2?

population mean

population variance

C)  Take a random sample of size n = 2 (with replacement). Determine the sampling distribution of the sample mean x̄, i.e. find the probability for each possible value of x̄.

sample size n=2 with replacement

population size N=3

No.of sample =

Sample unit	Sample mean (Yi)		probability
(1,1)	1	1	1/9
(1,2)	1.5	0.25	1/9
(1,3)	2	0	1/9
(2,1)	1.5	0.25	1/9
(2,2)	2	0	1/9
(2,3)	2.5	0.25	1/9
(3,1)	2	0	1/9
(3,2)	2.5	0.25	1/9
(3,3)	3	1	1/9
		3

sampling distribution of sample mean is

is unbiased estimate of population mean

sample variance

Probability of each possible value of is 1/9

D. Find the standard error of x̄. An observed sample consists of 1 and 2. Based on this piece of information, what is the (estimated) standard error?

Standard error  

Observed sample is (1,2)