Question

In: Statistics and Probability

The assets​ (in billions of​ dollars) of the four wealthiest people in a particular country are...

The assets​ (in billions of​ dollars) of the four wealthiest people in a particular country are

26, 25, 22 ,13

Assume that samples of size

n=2

are randomly selected with replacement from this population of four values.

a. After identifying the 16 different possible samples and finding the mean of each​ sample, construct a table representing the sampling distribution of the sample mean. In the​table, values of the sample mean that are the same have been combined.

x overbarx

Probability

x overbarx

Probability

26

nothing

22

nothing

25.5

nothing

19.5

nothing

25

nothing

19

nothing

24

nothing

17.5

nothing

23.5

nothing

13

nothing

​(Type integers or​ fractions.)

b. Compare the mean of the population to the mean of the sampling distribution of the sample mean.

The mean of the​ population,

nothing​,

is

equal to

greater than

less than

the mean of the sample​ means,

nothing.

​(Round to two decimal places as​ needed.)

c. Do the sample means target the value of the population​ mean? In​ general, do sample means make good estimates of population​ means? Why or why​ not?

The sample means

do not target

target

the population mean. In​ general, sample means

do

do not

make good estimates of population means because the mean is

an unbiased

a biased

estimator.

Solutions

Expert Solution

Population mean, = (26+25+22+13)/4 = 21.5

Number of sample using sample size 2 = 42 = 16

Number Sample pair Sample Mean
1 (26, 26) 26
2 (26, 25) 25.5
3 (26, 22) 24
4 (26, 13) 19.5
5 (25, 26) 25.5
6 (25, 25) 25
7 (25, 22) 23.5
8 (25, 13) 19
9 (22, 26) 24
10 (22, 25) 23.5
11 (22, 22) 22
12 (22, 13) 17.5
13 (13, 26) 19.5
14 (13, 25) 19
15 (13, 22) 17.5
16 (13, 13) 13

Sample mean of the sampling distribution:

a)

Sample mean Frequency Probability
26 1 1/16 = 0.0625
25.5 2 0.125
25 1 0.0625
24 2 0.125
23.5 2 0.125
22 1 0.0625
19.5 2 0.125
19 2 0.125
17.5 2 0.125
13 1 0.0625

b)  The mean of the​ population, 21.5, is equal to the mean of the sample​ means, 21.5.

c) The sample means target the population mean. In​ general, sample means do make good estimates of population means because the mean is an unbiased estimator.


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