Question

Consider the following population: {7, 8, 8, 9, 9}. The value of μ is 8.2, but suppose that this is not known to an investigator. Three possible statistics for estimating μ are

Statistic 1: the sample mean, x

Statistic 2: the sample median

Statistic 3: the average of the largest and the smallest values in the sample

A random sample of size 3 will be selected without replacement. Provided that we disregard the order in which the observations are selected, there are 10 possible samples that might result (writing 8and 8*, 9 and 9* to distinguish the two 8's and the two 9's in the population.)

7, 8, 8*	7, 8, 9	7, 8, 9*	7, 8*, 9	7, 8*, 9*
7, 9, 9*	8, 8*, 9	8, 8*, 9*	8, 9, 9*	8*, 9, 9*

(a)

For each of these 10 samples, compute Statistics 1, 2, and 3.

	Statistic 1	Statistic 2	Statistic 3
7, 8, 8*
7, 8, 9
7, 8, 9*
7, 8*, 9
7, 8*, 9*
7, 9, 9*
8, 8*, 9
8, 8*, 9*
8, 9, 9*
8*, 9, 9*

(b)

Construct the sampling distribution of each of these statistics. (Enter each statistic from smallest to largest.)

Statistic 1
p(Statistic 1)

Statistic 2
p(Statistic 2)

Statistic 3
p(Statistic 3)

(c)

Which statistic would you recommend for estimating μ and why?

Using the sampling distributions above, the means of the three statistics are calculated to be E(Statistic 1) =  , E(Statistic 2) =  , and E(Statistic 3) =  . Since the mean of Statistic  ? 1 2 3 is closest to μ = 8.2, we pick the  ---Select--- sample mean sample median average of the largest and smallest values in the sample as the best of the three statistics for estimating μ.

Answer 1

a)

	Statistic 1	statistic 2	Statistic 3
7,8,8*	7.666667	8	7.5
7,8,9	8	8	8
7,8,9*	8	8	8
7,8*,9	8	8	8
7,8*,9*	8	8	8
7,9,9*	8.33	9	8
8,8*,9	8.33	8	8.5
8,8*,9*	8.33	8	8.5
8,9,9*	8.67	9	8.5
8*,9,9*	8.67	9	8.5

b)  

Statistic 1	7.67	8	8.33	8.67
Probability	1/10=0.1	4/10=0.4	3/10=0.3	2/10=0.2
Statistic 2	8	9
probability	7/10=0.7	3/10=0.3
Statistic 3	7.5	8	8.5
probability	1/10=0.1	5/10=0.5	4/10=0.4

c) E ( Statistic 1 ) =

= 7.67 * 0.1 + 8 * 0.4 + 8.33 * 0.3 + 8.67 * 0.2

= 8.2

E ( Statistic 2 ) =

= 8 * 0.7 + 9 * 0.3 = 8.3

E ( statistic 3 ) =

= 8.15

We have population mean =  μ = 8.2

Since the mean of Statistic 1 is closest to μ = 8.2, we pick the sample mean as the best of the three statistics for estimating μ.