HOMEWORK 1

This assignment is designed to illustrate how a software packagesuch as Microsoft Excel supplemented by an add-in such as PHStatcan enable one to calculate minimum sample sizes necessary in orderto construct confidence intervals for both population means andproportions and to construct these types of confidence intervals.You should use PHStat in order to accomplish all parts of thisassignment. You should not only find the required information, butyou should explain the meanings of your results for each problemand part of each problem in the context of the problem. You alsoshould provide business implications of the results at which youarrive for one part of either problems two and three and forproblem five.

Scenario of the Problem:

1. You have been asked by a certain political party to study themean age of the supporters of a certain candidate who is runningfor public office in an upcoming election. A random sample of thosewho have demonstrated their support for the candidate will bechosen in order to accomplish the desired study. In order toprovide estimates of the population mean age of the supporters ofthis candidate, what minimum sample sizes will be necessary underthe followingconditions?                          

1. The estimate desired will need to be computed with 98%confidence to within ±2 years when it is felt that the populationstandard deviation in the ages of the supporters of the candidateis 7.5 years.           
2. The estimate desired will now need to be computed with 95%confidence to within ±2 years when the population standarddeviation is 7.5years.                                                                          
3. The estimate desired will now need to be computed with 98%confidence to within ±2 years when the population standarddeviation is 6years.                                                                              
4. The estimate desired will now need to be computed with 98%confidence to within ±3 years when the population standarddeviation is 7.5 years.                                                                         

In your memo, be sure to comment onthe differences found in the calculation of the minimum samplesizes in the various parts of the above problem. Explain whydifferences in your answers exist. In doing so, make allcomparisons relative to the answer found in the first part of theproblem.                                                                                        

2. You now need to construct a confidence interval for the meanage of the supporters of the candidate. You select a random sampleof 80 identified supporters of the candidate. You find that theirmean age is 44.57 years. You believe that the population standarddeviation of the ages of the supporters of the candidate is 7.5years. Construct both 98% and 95% confidence intervals for the meanage of the supporters of the candidate. In your explanation,comment upon the effect of the change in confidence level on thewidth of your interval.

3. You no longer believe that the population standard deviation inthe ages of the supporters of the candidate is a known quantity.You therefore will use the sample standard deviation of the ages ofthe supporters as an estimate of this unknown population standarddeviation. You collect data from a random sample of supporters ofthe candidate. The data identifies the ages of a sample of thesupporters of the candidate. This data is shown in appendix onebelow. Construct both 98% and 95% confidence intervals for the meanage of the supporters of the candidate for this situation. At eachconfidence level, comment upon the change in the results of thisproblem 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 ofsupporters of the candidate that are usually loyal supporters ofthe political party that this candidate represents based upon theirattesting to this fact and their previous voting record. Whatminimum sample sizes will be necessary in order to estimate thedesired population proportion under the following conditions?

1. The estimate is desired to within ±8% with 98% confidence whenthe population proportion of supporters of the party is thought toequal 80%.
2. The estimate is desired to within ±8% with 98% confidence whenthe population proportion of supporters of the party isunknown.
3. The estimate is desired to within ±8% with 98% confidence whenthe population proportion of supporters of the party is thought toequal 95%.

Comment on the changes in the minimumsample sizes you have computed based upon the changes in theinformation given in the three parts of this problem.

5. You now need to estimate with 98% confidence the populationproportion of supporters of the candidate that describes itself asloyal to the political party represented by the candidate. Yourandomly sample the population of supporters of the candidate andascertain whether each one has been a loyal party supporter. Theresults of that sampling process are shown in appendix two below.Using this information, construct the required confidenceinterval.

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