Question

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). We then calculate the mean of the sample in order to estimate the mean of the population. What is the probability that our error (the distance between the sample statistic and the population parameter) is one apple or less?

Answer 1

below is sampling distribution of mean:

x1	x2	sample mean
3	3	3
5	3	4
11	3	7
1	3	2
3	5	4
5	5	5
11	5	8
1	5	3
3	11	7
5	11	8
11	11	11
1	11	6
3	1	2
5	1	3
11	1	6
1	1	1

population mean =(3+5+11+1)/4 =5

number of total samples =16

number of such samples where sample mean is within 1 of population mean =5 (since 5 samples have sample mean from 4 to 6)

therefore probability =5/16