Download Chapter 7 Worksheet 1 Sampling Distribution of the Mean

GBS 221
Trollen
Name_____________
Chapter 7 Worksheet 1
Sampling Distribution of the Mean
NOTE:
Work in pencil. You may need to make changes as we go over this in class.
Suppose you have the following population: {2, 3, 4, 5, 8, 11}
µ=______
σ=______
1)
What is this population's:
N=______
2)
Suppose you select a sample of size n=2 from this population. You might get the 2 and the 3 or maybe the 2 and
the 4. In the table below, complete the list of all possible samples of size n=2 and their sample means:
TABLE 1
items in sample
X
2.5
3.0
2, 3
2, 4
3)
What is the probability of selecting the 2 and the 3?_____. What is the probability of selecting the 2 and the
4?_____. In fact, each of the sample combinations has what probability of being the one selected?_____
4)
Graph the possible values for X when n=2. Notice the axes are already labeled.
P( X )
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
1
Sampling Distribution of the Mean
2
3
4
5
↑ 6
µ=5.5
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7
8
9
10
X
CH07_worksheet_1.doc
5)
Is it possible to select a sample of size 2 from the above population and obtain a sample mean whose value is
close to the true value of the population mean?_____ Is it possible to select a biased sample whose mean
differs from the true value of the population mean?_____
6)
Suppose a sample of size n=5 is to be taken instead. List all the possible samples of size n=5 and their respective
sample means. [HINT: there will be 6.]
TABLE 2
items in sample
sample’s mean
7)
Each of these sample combinations has what probability of being the one selected?_____
8)
Graph the possible values for the sample mean when n=5.
P( X )
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
9)
1
2
3
4
5
↑ 6
µ=5.5
7
8
9
10
X
Is it possible to select a sample of size 5 from the above population and obtain a sample mean whose value is
close to the true value of the population mean?_____ Is it possible to select a biased sample whose mean
differs from the true value of the population mean?_____
10) The grand mean is defined as the mean of the possible values for the sample mean. Calculate the grand mean
now by computing the mean of the values in the sample's mean column in Table 2 above.
Sampling Distribution of the Mean
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CH07_worksheet_1.doc
11) Compute the standard deviation of the values in the sample's mean column in Table 2 above. Use the formula
for a population standard deviation and use the grand mean as the value for µ.
12) Suppose a sample of size n=6 is to be taken instead. List all the possible samples of size 6 and their respective
sample means. ([HINT: there will be only 1.]
TABLE 3
items in sample
sample’s mean
13) This sample combination has what probability of being selected?_____
14) Graph the possible values for the sample mean when n=6.
P( X )
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
1
2
3
4
5
↑ 6
µ=5.5
7
8
9
10
X
15) Is it possible to select a sample of size 6 from the above population and obtain a sample mean whose value is
close to the true value of the population mean?_____ Is it possible to select a biased sample whose mean
differs from the true value of the population mean?_____
16) Which sample size (n=2, n=5 or n=6) has a higher risk of sampling error? Explain.
Sampling Distribution of the Mean
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CH07_worksheet_1.doc
Now, suppose you have the following population: {4, 4, 4, 4, 4, 4}
µ=______
17) What is this population's:
σ=______
N=______
18) List all samples of size n=5 from this population and compute their respective sample means (there will be 6):
TABLE 4
items in sample
sample’s mean
19) Each of these sample combinations has what probability of being the one selected?_____
20) Graph the possible values for the sample mean when n=5.
P( X )
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
1
2
3
4
↑
µ=4
5
6
7
8
9
10
X
21) Is it possible to take a sample of size n=5 from this population and compute a sample mean whose value is close
to the true value of the population mean?_____ Is it possible to get a biased sample whose mean is different
from the true value of the population mean?_____
22) Complete the following generalizations regarding the risk of sampling error (ROSE) when using a sample mean
to estimate the population mean:
♦
as the sample size increases, the ROSE _______.
♦
as the amount of dispersion in the population increases, the ROSE ________.
Sampling Distribution of the Mean
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CH07_worksheet_1.doc