In: Math
HOMEWORK 1
This assignment is designed to illustrate how a software package
such as Microsoft Excel...
HOMEWORK 1
This assignment is designed to illustrate how a software package
such as Microsoft Excel supplemented by an add-in such as PHStat
can enable one to calculate minimum sample sizes necessary in order
to construct confidence intervals for both population means and
proportions and to construct these types of confidence intervals.
You should use PHStat in order to accomplish all parts of this
assignment. You should not only find the required information, but
you should explain the meanings of your results for each problem
and part of each problem in the context of the problem. You also
should provide business implications of the results at which you
arrive for one part of either problems two and three and for
problem five.
Scenario of the Problem:
- You have been asked by a certain political party to study the
mean age of the supporters of a certain candidate who is running
for public office in an upcoming election. A random sample of those
who have demonstrated their support for the candidate will be
chosen in order to accomplish the desired study. In order to
provide estimates of the population mean age of the supporters of
this candidate, what minimum sample sizes will be necessary under
the following
conditions?
- The estimate desired will need to be computed with 98%
confidence to within ±2 years when it is felt that the population
standard deviation in the ages of the supporters of the candidate
is 7.5 years.
- The estimate desired will now need to be computed with 95%
confidence to within ±2 years when the population standard
deviation is 7.5
years.
- The estimate desired will now need to be computed with 98%
confidence to within ±2 years when the population standard
deviation is 6
years.
- The estimate desired will now need to be computed with 98%
confidence to within ±3 years when the population standard
deviation is 7.5 years.
In your memo, be sure to comment on the differences found in the
calculation of the minimum sample sizes in the various parts of the
above problem. Explain why differences in your answers exist. In
doing so, make all comparisons relative to the answer found in the
first part of the
problem.
- You now need to construct a confidence interval for the mean
age of the supporters of the candidate. You select a random sample
of 80 identified supporters of the candidate. You find that their
mean age is 44.57 years. You believe that the population standard
deviation of the ages of the supporters of the candidate is 7.5
years. Construct both 98% and 95% confidence intervals for the mean
age of the supporters of the candidate. In your explanation,
comment upon the effect of the change in confidence level on the
width of your interval.
- You no longer believe that the population standard deviation in
the ages of the supporters of the candidate is a known quantity.
You therefore will use the sample standard deviation of the ages of
the supporters as an estimate of this unknown population standard
deviation. You collect data from a random sample of supporters of
the candidate. The data identifies the ages of a sample of the
supporters of the candidate. This data is shown in appendix one
below. Construct both 98% and 95% confidence intervals for the mean
age of the supporters of the candidate for this situation. At each
confidence level, comment upon the change in the results of this
problem from the results of the previous problem.
Appendix One: (Age of Supporters)
40
32
60
58
22
28
66
70
71
55
59
58
62
44
89
48
56
33
46
39
39
44
32
48
49
50
51
18
28
23
34
54
28
76
35
77
38
21
59
51
54
38
45
39
19
90
37
46
22
26
27
39
30
45
27
- You also need to estimate the population proportion of
supporters of the candidate that are usually loyal supporters of
the political party that this candidate represents based upon their
attesting to this fact and their previous voting record. What
minimum sample sizes will be necessary in order to estimate the
desired population proportion under the following conditions?
- The estimate is desired to within ±8% with 98% confidence when
the population proportion of supporters of the party is thought to
equal 80%.
- The estimate is desired to within ±8% with 98% confidence when
the population proportion of supporters of the party is
unknown.
- The estimate is desired to within ±8% with 98% confidence when
the population proportion of supporters of the party is thought to
equal 95%.
Comment on the changes in the minimum sample sizes you have
computed based upon the changes in the information given in the
three parts of this problem.
- You now need to estimate with 98% confidence the population
proportion of supporters of the candidate that describes itself as
loyal to the political party represented by the candidate. You
randomly sample the population of supporters of the candidate and
ascertain whether each one has been a loyal party supporter. The
results of that sampling process are shown in appendix two below.
Using this information, construct the required confidence
interval.
Appendix Two: (Loyal Party Supporter? (Y = yes, N = no))
Y
Y
Y
Y
N
N
Y
Y
Y
Y N
Y
N
Y
Y
Y
Y
Y
Y
Y
N Y
N
Y
Y
Y
Y
Y
Y
Y
Y
Y Y
Y
Y
N
N
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
N
Y
N
N
Y
N
Y
Y
Y
Y
Y
Y
Y
Y
N Y
Y
Y
Y
N
Y
Y
Y
Y
Y
Y N
N
Y
Y
Y
Y
Y
Y
Y
Y
Y Y