1. Three randomly selected households are surveyed. The numbers of people in the households are 3, 4 and 11. Assume that samples of size n=2 are randomly selected with replacement from the populatio…
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Question “1. Three randomly selected households are surveyed. The numbers of people in the households are 3, 4 and 11. Assume that samples of size n=2 are randomly selected with replacement from the populatio…”
1. Three randomly selected households are surveyed. The numbers
of people in the households are 3, 4 and 11. Assume that samples
of size n=2 are randomly selected with replacement from the
population of
3, 4, and 11. Listed below are the nine different samples.
Complete parts (a) through (c).
3,3 3,4 3,11 4,3 4,4 4,11 11,3 11,4 11,11
a. Find the variance of each of the nine
samples, then summarize the sampling distribution of the variances
in the format of a table representing the probability distribution
of the distinct variance values.
s2 | Probability |
b. Compare the population variance to the mean
of the sample variances.
c. Do the sample variances target the value of
the population variance? In general, do sample variances make
good estimators of population variances? Why or why not?
2. Assume a population of 1, 4, and 10. Assume that samples of
size n=2 are randomly selected with replacement from the
population. Listed below are the nine different samples. Complete
parts a through d below.
1,1
1,4
1,10
4,1
4,4
4,10
10,1
10,4 10,10
a. Find the value of the population standard
deviation.
b. Find the standard deviation of each of the
nine samples, then summarize the sampling distribution of the
standard deviations in the format of a table representing the
probability distribution.
c. Find the mean of the sampling distribution
of the sample standard deviations. The mean of the sample standard
deviations is the sum of the nine sample standard deviations
divided by the number of samples.
d. Do the sample standard deviations target the
value of the population standard deviation? Ingeneral, do sample
standard deviations make good estimators of population standard
deviations? Why or why not?
3. Three randomly selected households are surveyed. The numbers
of people in the households are 3, 4, and 11. Assume that samples
of size n=2 are randomly selected with replacement from the
population of 3, 4, and 11.Construct a probability distribution
table that describes the sampling distribution of the proportion of
odd numbers when samples of sizes n=2 are randomly selected. Does
the mean of the sample proportions equal the proportion of odd
numbers in the population? Do the sample proportions target the
value of the populationproportion? Does the sample proportion make
a good estimator of the population proportion? Listed below are
the nine possible samples.
3,3 3,4 3,11 4,3 4,4 4,11 11,3 11,4 11,11
Construct the probability distribution table.
Sample ProportionProbability |
4. The assets (in billions of dollars) of the four wealthiest
people in a particular country are 43, 28, 25, 16. 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.
b. Compare the mean of the population to the mean of the
sampling distribution of the sample mean.
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?
Answer
2a)
Each sample is equally probable, so the probability of getting any sample you request is 1/9.
Each sample’s variance is
Var
Here, X1 and 2 show the observations of the samples. The following table lists the probabilities and variances for each sample:
The following table illustrates the probability distribution for sample variances.
b)
The following table shows how to calculate the mean population variances.
The mean of the sample variances is therefore
Variation in population is
The variance of the population is equal to the mean variances from the sample.
c)
Sample variances are used to estimate the population variance. Sample variances are good at estimating population variances.
=\frac{(x 1-\text { mean })^{2}+(x 2-\text { mean })^{2}}{2-1}=(x 1-\text { mean })^{2}+(x 2-\text { mean })^{2}
Samples
Variances,s^2
P(s^2)
3
3
0
0.11111111
3
4
0.5
0.11111111
3
11
32
0.11111111
4
3
0.5
0.11111111
4
4
0
0.11111111
4
11
24.5
0.11111111
11
3
32
0.11111111
11
4
24.5
0.11111111
11
11
0
0.11111111
Variances,s^2
P(s^2)
0
0.3333
0.5
0.2222
24.5
0.2222
32
0.2222
Variances,s^2
P(s^2)
s^2P(s^2)
0
0.3333
0
0.5
0.2222
0.1111
24.5
0.2222
5.4439
32
0.2222
7.1104
Total
12.6654
E\left(s^{2}\right)=\sum s^{2} P\left(s^{2}\right)=12.6654
\sigma^{2}=12.6667
Conclusion
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