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 thetable, 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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