Kevin Tillery

                                                                                                                         Matt Schaefer

                                                                                                                         Jena Everhard

                                                                                                                               5/19/2003

                                                                                                                           AP Statistics

 

 

Disclaimer:  “This study was done in an AP Statistics course with relatively small sample sizes.  The validity of such studies must always be questioned.  Please keep this in mind if you use or report the results of this study.”

           

Abstract: In this observational study, a group of both fifteen males and females were randomly selected from an acquired list of registered drivers, male and female (population), at a local high school.  From these two groups it became possible to answer a question involving the type of vehicle each group drove to school.  Specifically, the question involves the difference in proportion between both males and females, which drove domestic cars to TCHS.  Clinging to a hypothesis that each group should hold relatively the same proportion of domestic cars driven to school, however it is possible otherwise that each group are in fact quite different in proportion.  At first thought it simply just seems unlikely that the two groups should differ due to any outside variables.  Any discrepancies among what kind of cars each group presumably drive should be answered and so by performing inference all questions should be clarified.  The two-proportion z test makes it possible to compare each SRS from both the male and female populations.

            After the assumptions were proved to correct, the test was run.  A verdict of failing to reject surfaces as an unusually high p-value is calculated.  The p-value of .712 forces upon this analysis as an alpha level below .05 is needed to prove any evidence statistically significant, therefore rejecting the theory that the two categories aren’t associated is irrelevant.  The two categories actually are associated.

 

Methodology:  The question we are trying to answer is whether or not the proportion of males that drive domestic vehicles to TCHS differs from the proportion of females that commute to TCHS in a domestic vehicle.  By numbering an acquired population list of all student drivers it was possible to take an SRS of fifteen males and females.  From here both randomly selected groups of fifteen were asked the make of their vehicle and the results were tallied.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Exploratory Data Analysis:

 

 
 
 

 

 

 

 

 

 

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Pie Chart: C1

 

 

Test and CI for Two Proportions

 

Sample      X      N  Sample p

1           8     15  0.533333

2           9     15  0.600000

 

Estimate for p(1) - p(2):  -0.0666667

95% CI for p(1) - p(2):  (-0.420507, 0.287174)

Test for p(1) - p(2) = 0 (vs not = 0):  Z = -0.37  P-Value = 0.712

 

 

 

 

 

Analysis and Inference/Conclusion:  The assumptions must be addressed first and foremost before moving on.  For this particular test the data must be an SRS from the population of interest.  Next, the population needs to be ten times the size of the sample.  Lastly, both sample sizes have to be ten or more.  Luckily, the data does indeed fit these criteria.
            From here the alpha level or basically the number which proves the test to be significant must be determined.  Since there is no involvement with the medical community normally needing a lower alpha level, an alpha level slightly higher is chosen of .05.  After determining the alpha level, both Ho and Ha are selected.  For this particular problem Ho involves proportion one and two being equal.  Thus the opposite for Ha is that each proportion is not equal.

            Calculating the Z statistic makes it possible to find the p-value by using table A.  The Z statistic is calculated through a simple formula.  To carryout this formula p-hat two is subtracted from p-hat one.  P-hat is the number of successes over the total count of both successes and failures.  This is performed on each population.  Subtracting the two gives the number of -.067.  This number is on top of the rest of the equation.

            On the bottom is also p-hat.  This one however is different from the other two.  For this p-hat both successes from each group are added together and divided by thirty.  The number found is .5667.  Then this number is multiplied by one minus itself.  Accomplishing this task, moving on to the next involves adding one over fifteen twice, then multiplying it by the number found in the previous task.  This time the number is .03274.  Taking the square root gives the number .18094 on the bottom.  Fairly reasonably dividing the numerator by the denominator the number -.37 emerges as the Z-statistic.  Table A in the front of the book shows a p-value of .3557.  Multiplying this by two gives the true p-value of .712.

            With such a high p-value considerably over the original alpha level of .05, Ho is failed to be rejected.  There is a seventy percent chance that the results found didn’t happen by chance.  Indeed both population proportions of domestic cars are equal and so the hypothesis stated above is correct.  The evidence is not statistically significant and so Ho isn’t rejected.  The proportion of males to females actually is in roughly the same proportion.