Answer to Question #299740 in Statistics and Probability for Hannah

Question

. A population consists of the four numbers 1, 2, 8, and 9. List all the possible

samples of size n = 2 which can be drawn without replacement from the population.

Find the following:

a. Population mean

b. Population variance

c. Population standard deviation

d. Mean of the sampling distribution of sample means

e. Variance of the sampling distribution of sample means

f. Standard deviation of the sampling distribution of sample means

Answer 1

list of possible samples = (4C2) = 6 = ( (1,2), (1,8) , (1,9), (2,8),(2,9) ,(8,9) )

a) population mean = (1+2+8+9) /4 = 5

b) Population variance

= ( (1-5)² + (2-5)² + (8-5)²+ (9-5)² ) / (3)

= (16 + 9 + 9 +16) / 3

= 16.6667

c.) Population standard deviation = (Population variance)^1/2 = (16.667)^1/2 = 4.0825

d.) Mean of the sampling distribution of sample means

we obtain the sample means as below

(1+2)/2 =1.5

(1+8)/2 = 4.5

(1+9)/2 = 5

(2+8)/2 = 5

(2+9)/2= 5.5

(8+9)/2 = 8.5

Thus the overall mean = (1.5 + 4.5 + 5 + 5 + 5.5 + 8.5)/6 = 5

e.) Variance of the sampling distribution of sample means

it is defined as below

= ( (1.5² + 4.5² + 5² + 5² + 5.5² + 8.5²) / (6) ) - (5²)

= ( (175) / (6) ) - (25)

= 29.1667 - 25

= 4.1667

f.) Standard deviation of the sampling distribution of sample means

this equals to (Variance of the sampling distribution of sample means)^1/2 = (4.1667)^1/2

which yields 2.04125