1. 1. John Michael Croft Kennesaw State University Stat 3120-01 December 2, 2010 Final Project
2. 2. Introduction I want to begin with a proactive apology in advance for any lack of formality with regards to citing sources or references found throughout this paper. I feel complete formality unnecessary as I do not aspire to be the Bohr or Keynes of the statistical social science world and have no aspirations of this being published in any significant journals. I have attached two appendices: the first of which will contain all charts, figured, graphs, etc. that shall be referenced throughout, and the second will contain any and all SAS code used to report my findings. I created one variable on my own; however I obtained the idea from another data set found on a similar www.businessweek.com site. The variable is labeled as SalperTuit and is constructed from taking the median salary reported and dividing it by the cost of one semester for the corresponding college or university. This variable is intended to be viewed as a type of return on investment (ROI) with regards to starting median salary with respect to the cost of one semester. I also have constructed another variable found later in this report labeled as TotCost which is the cost per semester multiplied by the length of the program. My findings reported below primarily consist of analyzing select variables, while comparing the differences between public and private school graduates. The primary variables throughout my analyses are: School Type, Job Placement Grade, Teacher Quality Grade, Median Starting Salary, Length of Program, Cost per Semester, Total Cost, Salary per Tuition Dollar per Semester, and ROI Rank. With regards to any lack of formality, I want to state the website in which I obtained my data from as far in advance as possible so not to mislead the reader as to the originality of the data that has been analyzed. The data set consist of the top 112 business undergrads schools in the United States and can be obtained at: http://www.businessweek.com/interactive_reports/bschools_undergraduate_10rankings. html This topic (public and private schools) has been of interest to me for a few years now. However it became of significant interest while I was interning for the Department of Justice in the summer of 2009 when the girl I was dating told me it cost her (or Im sure her parents) $50,000 per year for tuition. Ive also completed a similar analysis in the past on the top Research Universities and Liberal Arts Colleges located in the United States.
3. 3. Research Hypotheses The main purpose of this analysis is to show, hopefully, that private schools are overrated in relation to median starting salary. This is to say that while the difference in starting salaries for private school graduates may be significantly higher, its not so much higher as to justify the significantly higher tuition cost. And actually I feel the higher median starting salary one receives from attending a private school is nominal, and almost unwarranted, as compared to the significantly higher tuition cost. Below, I compare several variables based on type of schools, public or private, to help determine what encourages students to attend private schools over public schools. I also look at what helps to determine starting median salary between different school types. Descriptive Statistics Qualitative Variables Figures: 29 and 30 are pie charts of percent frequencies for Teacher Quality Grade and Job Placement Grade. One should be aware the distributions appears identical with the mode of both being a grade of B. According to Figure: 1 with respect to job placement grade, 19.82% or 22 schools have a grade of A+, 25.23% or 28 schools have a grade of A, 35.14% or 39 schools have a grade of B, and 19.82% or 22 schools have a grade of C. Grade B is the mode (the grade that occurs the most). According to Figure: 3 with respect to job placement grade, 14.29% or 9 public schools have a grade of A+, 26.98% or 17 public schools have a grade of A, 39.68% or 25 public schools have a grade of B, and 19.05% or 12 public schools have a grade of C. Again, grade B is the mode. According to Figure: 5 with respect to job placement grade, 27.08% or 13 private schools have a grade of A+, 22.92% or 11 private schools have a grade of A, 29.17% or 14 private schools have a grade of B, and 20.83% or 10 private schools have a grade of C. And again, grade B is the mode. According to Figure: 7 with respect to teacher quality grade, 19.82% or 22 schools have a grade of A+, 25.23% or 28 schools have a grade of A, 35.14% or 39 schools have a grade of B, and 19.82% or 22 schools have a grade of C. Grade B is the mode. According to Figure: 9 with respect to teacher quality grade, 14.29% or 9 public schools have a grade of A+, 26.98% or 17 public schools have a grade of A, 39.68% or 25 public schools have a grade of B, and 19.05% or 12 public schools have a grade of C. And again, grade B is the mode. According to Figure: 11 with respect to teacher quality grade, 27.08% or 13 private schools have a grade of A+, 22.92% or 11 private schools have a grade of A, 29.17% or 14 private schools have a grade of B, and 20.83% or 10 private schools have a grade of C. And again, grade B is the mode.
4. 4. Quantitative Variables Figure 21 shows the descriptive statistics for the quantitative variables analyzed for this report. Median starting salary has a mean of $48,310 and a median of $48,500 with a minimum of $36,000 and a maximum of $62,000 as well as a standard deviation of $5650.05. Starting salary per one tuition dollar per semester has a mean of $4.05 and a median of $4.14 with a minimum of $1.13 and a maximum of $11.70 as well as a standard deviation of $2.74. Cost per semester has a mean of $19,882.39 and a median of $11,732 with a minimum of $3,738 and a maximum of $41,610 as well as a standard deviation of $13,605. Length of program has a mean of 3.58 years and a median of 4 years with a minimum of 2 years and a maximum of 4 years as well as a standard deviation of .803 years. Total program cost has a mean of $74,470 and a median of $46,928 with a minimum of $7,476 and a maximum of $166,440 as well as a standard deviation of $55,632. ROI rank has a mean and median of 56 with a minimum of 1 and a maximum of 111 as well as a standard deviation of 32.19 places. Figure 22 shows descriptive statistics for the same quantitative variables discussed above from Figure 21, however they are broken down by school type. I would like to point out some seemingly significant aspects of Figure 22; however I encourage the reader to take a complete look in appendix I if interested above and beyond what I report here. On average, the starting salary per tuition dollar per semester for public schools is $4.22 higher than private schools. On average, private schools cost $25,710 more per semester than public schools. On average, private school total program cost is approximately $102,000 more than public schools. And on average, the ROI rank is 53 spots lower for public schools over private schools with lower rankings being better. Figure 31 is a bar chart of starting median salary. The distribution appears to be normal with the mean being a preferred benchmark over the median. Figure 35 is a bar chart of starting median salary by type of school. The distribution for each appears to be normal with the mean being a preferred benchmark over the median. Private school students have a higher maximum while public school students have a lower minimum, however the mean of both appear to be similar. Figure 32 is a bar chart of starting salary per tuition dollar spent per semester. The distribution appears to not be normal but seems to be decreasing at a decreasing rate, on average. The median would be a better benchmark for starting salary per tuition dollar per semester. Figure 36 is a bar chart of starting salary per tuition dollar per semester by school type. It shows that private school students have a fairly low starting salary per tuition dollar spent per semester when compared to that of public school students.
5. 5. Figure 33 is a bar chart of tuition cost per semester. The distribution appears to not be normal and appears to be bimodal with the modes appearing towards the minimum and maximum of the distribution. The median would be a better benchmark for tuition cost per semester. Figure 37 is a bar chart of tuition cost per semester by school type. It shows that private school students appear to pay a much more in tuition as compared with public school students. With the exception of a small percent of private schools, it appears the maximum cost for public schools is lower than the minimum for a private school. Figure 34 is a bar chart of total tuition cost. The distribution appears to not be normal and appears to be bimodal with the modes appearing towards the minimum and maximum of the distribution. The median would be a better benchmark for tuition cost per semester. Figure 38 is a bar chart of total tuition cost by school type. It shows that private school students appear to pay a much more in tuition as compared with public school students. Again, with the exception of a small percent of private schools, it appears the maximum cost for public schools is lower than the minimum for a private school. Results From the previous section, we notice, on average, that public and private undergrad schools have grades of B for job placement grade and teacher quality. Again, I would like to point out that the results for job placement grade and teacher quality grade appear identical and may be highly correlated. According to Figure: 2, we are 95% confident the true population proportion of top business undergrad schools that have a job placement grade of A+ is between 10.07% and 29.57%. According to Figure: 4, we are 95% confident the true population proportion of top business undergrad public schools that have a job placement grade of A+ is between 2.93% and 25.64%. According to Figure: 6, we are 95% confident the true population proportion of top business undergrad private schools that have a job placement grade of A+ is between 10.56% and 43.61%. From figure 13, we are 95% confident the true population proportion difference between private school and public schools with job placement grades of A or higher is not significantly different since the interval runs from -9.52% and 27.38% in which 0% falls between. According to Figure: 8, we are 95% confident the true population proportion of top business undergrad schools that have a teacher quality grade of A+ is between 10.07% and 29.57%. According to Figure: 10, we are 95% confident the true population proportion of top business undergrad public schools that have a teacher quality grade of A+ is between 2.93% and 25.64%. According to Figure:
6. 6. 12, we are 95% confident the true population proportion of top business undergrad private schools that have a teacher quality grade of A+ is between 10.56% and 43.61%. From figure 14, we are 95% confident true population difference between private school and public schools with teacher quality grades of A or higher is significantly different, with a difference between 36.80% and 68.76% with private schools being more likely to have higher grades. From Figures 15-20, we see the 95% confidence intervals for the quantitative variables analyzed for this report, all of which appear to be significant with respect to the top undergrad business school by school type (public or private), as well as the difference between the types which all also appear to be significant. According to Figure: 15, we are 95% confident the true population mean median starting salary for private schools is between $48,585 and $51,899 with a mean of $50,242 compared to $45,534 and $48,145 with a mean of $46,840 for public schools. Furthermore we see the true population difference of public schools and private schools is between $1316 and $5489 with an average difference of $3402 where private schools having higher starting salaries. According to Figure: 16, we are 95% confident the true population mean cost per semester for private schools is between $32,650 and $36,300 with a mean of $34,474 compared to $8,052 and $9,477 with a mean of $8,764 for public schools. Furthermore we see the true population difference of public schools and private schools is between $23,762 and $27,659 with an average difference of $25,710 where private schools are more expensive. According to Figure: 17, we are 95% confident the true population mean program length for private schools is between 3.65 and 5 years with a mean of 3.81 years compared to 3.17 and 3.63 years with a mean of 3.40 years for public schools. Furthermore we see the true population difference of public schools and private schools is between .13 and .69 years with an average difference of .42 years where private schools take longer to complete. According to Figure: 18, we are 95% confident the true population mean starting salary per one tuition dollar per one semester for private schools is between $1.23 and $2.09 with a mean of $1.66 compared to $5.38 and $6.37 with a mean of $5.88 for public schools. Furthermore we see the true population difference of public schools and private schools is between $3.57 and $4.87 with an average difference of $4.22 with public schools being paid more per tuition dollar spent per semester.
7. 7. According to Figure: 19, we are 95% confident the true population mean ROI rank for private schools is between 80.87 and 91.54 with a mean of 86.21 compared to 28.36 and 37.61 with a mean of 32.98 for public schools. Furthermore we see the true population difference of public schools and private schools is between 46.24 and 60.21 places with an average difference of 53.22 places where public schools have better ROI Rankings (given a lower ranking is better). According to Figure: 20, we are 95% confident the true population mean difference of Total Tuition and Starting Median Salary for private schools is between $73,807 and $90,752 with a mean of $82,279 (meaning on average private school students spend $82,279 more on tuition than they make in the first year of employment) compared to $13,078 and $20,117 with a mean of $16,598 for public schools (meaning on average public school students make $16,568 more the first year of employment than their total tuition). Furthermore we see the true population difference of public schools and private schools is between $89,756 and $107,998 with an average difference of $98,877 where public schools net, on average, $98,877 more than private schools, the first year of employment. From Figure 23, we see that best linear model to explain starting median salary as explained by cost per semester is: Med_Sal = 45,118 + .1606(Cost) On average students will start out making $45,118 with an additional $16.06 per $100 spent on tuition per semester. Both the intercept and the slope parameter estimates appear to be significant, with 14.17% of the variation in Starting Median Salary being explained by our model. However, I stop short of saying this is the best model possible as a linear relationship may not be the best representation. Figure 24 is a scatter plot of starting median salary and cost of tuition per semester. There does appear to be a general increase in median salary as cost per semester increases, however, again there may not be a linear relationship. Starting median salary appears to increase initially as cost per semester increases, however it then flattens out, begins to decrease and then resumes increasing. A cubic model may be a better fit to explain the relationship show in this figure. I also want to say this may even be a discontinuous graph. Figure 25 shows a scatter plot of cost (as the independent variable) plotted with the residual value from predicting starting median salary. The residuals appear to be concentrated towards the bottom and towards the top of the scatter plot with an obvious gap in the middle. The gap appears to separate the public schools from the private schools. Further analyses would need to be conducted to make any meaningful conclusions to this figure.
8. 8. From Figure 26, we see that best linear model to explain starting salary per tuition dollar per semester as explained by cost per semester is: SalperTuit = 7.552 - .0001759(Cost) On average students will start out making $7.55 per tuition dollar per semester spent minus $.1759 per $1,000 spent on tuition per semester. Both the intercept and the slope parameter estimates appear to be significant with our model explaining 76.38% of the variation in Sal per Tuition Dollar per Semester. However, this is probably not the best model possible as a linear relationship may not be the best way to model this data. Figure 27 shows a scatter plot between starting salary per tuition dollar per semester and cost per semester. I feel confident in saying that this relationship is not linear and further analyses are required to make meaningful statement about this relationship. The figure shows that starting salary per tuition dollar per semester is decreasing at a decreasing rate as cost per semester increase with some type of log linear relationship. Figure 28 is one of the more interesting graph as it show a hyperbolic relationship between cost per semester (as the independent variable) and the residuals from the prediction of salary per tuition dollar spent per semester. Further analyses are needed to make a meaningful conclusion, however the relationship does not appear normal or random. Conclusion In conclusion, I would like to point out some significant points of interested noted in the above sections. Compared to public schools, private school programs take, on average, almost six months longer to complete, and their students start out making $3402 more their first year of employment, however tuition cost $25, 710 more per semester while public schools grads making $4.22 more per tuition dollar spent per semester. For a four year program, it would cost a private school graduate over $100,000 more than a public school graduate and take approximately four years to recoup the tuition with a starting salary of $52,000 where the public school graduate recoups her tuition the first year. By the time a private school graduate recoups her tuition, a public school graduate will have netted approximately $157,000 more over a four year period. I also want to again point out the true population difference of public schools and private schools is between $89,756 and $107,998 with an average difference of $98,877 where public schools net, on average, $98,877 more than private schools, the first year of employment. The author is a graduate of a public school, thus making his opinion of private schools possibly somewhat bias, but I hope my findings somewhat justify this opinion.
9. 9. Further Research I want to encourage any and all possible further research on this topic as there several other variables that need to be analyzed in relation to median starting salary that have not been addressed in this report. If not for higher starting salaries, why do students choose to attend private schools? Some areas of interest may be to study the correlations and other relationships with: acceptance into better grad schools programs, time it takes to get a job after completion of undergrad program, ability to be more selective in who one works for, how early and often one might receive a raise, ability to learn from more knowledgeable professors. This is by no means an exhaustive list as there are several more one can chose to research.
10. 10. Appendix I Figure 1: Frequency Chart of Job Placement Grade Grade Frequency Percent Cumulative Frequency Cumulative Percent A+ 22 19.82 22 19.82 A 28 25.23 50 45.05 B 39 35.14 89 80.18 C 22 19.82 111 100.00 Figure 2: Z-Test for Job Placement Grade = A+ Proportion 0.1982 ASE 0.0378 95% Lower ConfLimit 0.1007 95% UpperConf Limit 0.2957 Exact ConfLimits 95% Lower Conf Limit 0.1110 95% UpperConf Limit 0.3122 Test of H0: Proportion = 0.5 ASE under H0 0.0475 Z -6.3594 One-sidedPr< Z |Z|