Question

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:

1. 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?                          

1. 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.            
2. 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.                                                                          
3. The estimate desired will now need to be computed with 98% confidence to within ±2 years when the population standard deviation is 6 years.                                                                              
4. 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.                                                                                        

2. 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.

3. 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       

4. 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?

1. The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is thought to equal 80%.
2. The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is unknown.
3. 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.

5. 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        

Answer 1

1) Sample size estimation

Margin of error E =

n =

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.        

alpha = 0.02

Z_0.01 = 2.33

n = = 77

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.            

alpha = 0.05

Z_0.025 = 1.96

n = 55

The estimate desired will now need to be computed with 98% confidence to within ±2 years when the population standard deviation is 6 years.             

n = 49

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

n = 34

2) Confidence intervals

Given sample mean = 44.57 years

sample size n = 80

population std dev = 7.5 years

CI =

98% CI = (44.57 -2.33*7.5/Sqrt(80) , 44.57 +2.33*7.5/Sqrt(80) ) = (42.616, 46.524)

95% CI = (44.57 -1.96*7.5/Sqrt(80) , 44.57 +1.96*7.5/Sqrt(80) ) = (42.926 , 46.214)

3)

we have to use t-test

CI = \bar{x} \pm t_{\alpha /2, n-1}*s/\sqrt{n}

t_0.01,54 =

N	Mean	StDev	SE Mean	98% CI for μ
55	45.00	17.27	2.33	(39.42, 50.58)

μ: mean of Data

N	Mean	StDev	SE Mean	95% CI for μ
55	45.00	17.27	2.33	(40.33, 49.67)

μ: mean of Data

4)

The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is thought to equal 80%.

n =

n= = 136

The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is unknown.

p =0.5

n = = 213

The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is thought to equal 95%.

n = = 41

5)

From the given sample number of Y = 71

total sample size = 88

p = 71/88 = 0.806

CI =

98% CI = (0.708913, 0.904723)

Descriptive Statistics

N	Event	Sample p	98% CI for p
88	71	0.806818	(0.708913, 0.904723)