Question

2-4 There is a population of four people: Amy, Ben, Clara, and Donna. [Note: this is a population, not a sample.] They are asked how many apples they ate last month. Amy ate 3 apples, Ben ate 5 apples, Clara ate 11 apples, and Donna ate 1 apple. We are too lazy to poll the entire population, so we take a sample of two (with replacement and order matters). What is the probability of the sample mean equaling the population mean?

Answer 1

Population is x: 3 ,5, 11, 1.

The mean of the population is Mu=(3+5+11+1)/4=5

The possible sample of size 2 with replacement from the above population and their mean is given below:

Sl No.	First sample	Second sample	Mean
1	3	3	3
2	5	5	5
3	11	11	11
4	1	1	1
5	3	5	4
6	3	11	7
7	3	1	2
8	5	3	4
9	11	3	7
10	1	3	2
11	5	11	8
12	5	1	3
13	11	5	8
14	1	5	3
15	11	1	6
16	1	11	6

From the list of 16 means, only one mean of the sample (5, 5) is 5. Hence, the probability of the sample means equal to the population mean is 1/16=0.0625.