Questions:
(a) Provide a descriptive analysis of the two variables (e.g., mean, standard deviation, minimum and maximum).
(b) Develop a scatter diagram with retention rate as the independent variable. What does the scatter diagram indicate about the relationship between the two variables?
(c) Develop and estimate a regression equation that can be used to predict the graduation rate (%) given the retention rate (%).
(d) State the estimated regression equation and interpret the meaning of the slope coefficient.
(e) Is there a statistically significant association between graduation rate (%) and retention rate (%). What is your conclusion?
(f) Did the regression equation provide a good fit? Explain.
(g) Suppose you were the president of South University. After reviewing the results, would you have any concerns about the performance of your university compared to other online universities?
(h) Suppose you were the president of the University of Phoenix. After reviewing the results, would you have any concerns about the performance of your university compared to other online universities?
Your report need to be structured as follows:
1. Purpose
2. Background
3. Method
4. Result
5. Discussion
6. Recommendation
It has been pointed out that during the last few years, the higher education sector in the USA has undergone through a significant alteration in its market place due to the momentous growth rate of online universities (Bekker and Kleibergen, 2001). Though certain growth is there, in terms of proportion of graduation rate as well as the proportion of retention rate has significantly fluctuated during this time frame. It is generally considered that higher the retention rate means the university will evidence higher graduation rate. Under such circumstances, any fluctuation in terms of graduation rate as well as retention rate turns out to be a major concern for the online universities (Kamath, 2009).
Now, if the recent economic trend has been considered here, then it can be found out that the organizations are experiencing cutthroat rivalry, whether they are belonging in a specified industry or stayed in different industry. The revelry exists here mainly because of the fact that they intended to go ahead in order to gain the competitive advantage (Kim and Chambers, 2011). Under such circumstances, an in depth economic analysis is required by the online education providing universities so that they will able to comprehend the recent market trend with considering the two major factors “graduation rate” and the “retention rate”.
Over the time frame as the online universities in the USA are experiencing significant fluctuation in terms of retention rate as well as gradation rate. Since, the retention rate significantly influence the proportion of graduation rate in each universities, it is become essential to understand such fluctuation in the both variables have any significant impact on overall performance of the online universities or not. Under such circumstances, this study is aimed to assess this particular scenario based on these to variables and for that a sample of 29 online universities in the United States were considered here. Data related to these two variables are assorted in terms of percentage. Here, particularly it is intended to assess whether the influence of the retention rate over the graduation rate for selected universities in against the target population has been changed or not.
Background
Since, the objective of the study is to measure the effect of retention rate over graduation rate for the selected online universities; it is become apparent that analysis through statistical tool will be appropriate here (HaÃŒË†rdle and Simar, 2012). So, it can be said that basically this study followed a descriptive research design, where mainly secondary data related to both the variables will be assessed here using statistical apparatus. Now, it is also the fact that execution of any statistical study primarily necessitates a descriptive analysis for the reason that it will help to employ inferential statistical tool in more effective manner.
Under such circumstances, a brief descriptive statistical analysis has been executed here using statistical parameters like mean, standard deviation, minimum and maximum (Gionis, 2013). Next, the study followed in depth inferential analysis using multiple regression models. In this context, it is noted that executing any statistical analysis both in terms of descriptive study or inferential study, various instruments like SPSS, Excel, etc were employed by the researcher. Here, the study mainly utilized the Microsoft Excel to do every calculation. Thus, it can be said that the study follows the below mentioned steps:
Step 1: Descriptive analysis of graduation rate and retention rate;
Step 2: Scatter diagram of graduation rate and retention rate;
Step 3: Multiple regression as inferential study;
The below mentioned section provides details of the results found following the above mentioned three steps.
Calculation of descriptive statistics:
Retention Rate (%) 
Graduation Rate (%) 

Mean 
57.41379 
Mean 
41.75862 
Standard Deviation 
23.24023 
Standard Deviation 
9.865724 
Minimum 
4 
Minimum 
25 
Maximum 
100 
Maximum 
61 
Count 
29 
Count 
29 
Table 1: Descriptive statistics
Scatter diagram of graduation rate and retention rate with retention rate as the independent variable:
Figure 1: Scatter diagram of retention rate and graduation rate
Multiple regression analysis:
Regression Statistics 

Multiple R 
0.670 
R Square 
0.449 
Adjusted R Square 
0.429 
Standard Error 
7.456 
Observations 
29 
Table 2: Regression statistics table
ANOVA 


df 
SS 
MS 
F 
Significance F 
Regression 
1 
1224.286 
1224.286 
22.022 
0.000 
Residual 
27 
1501.024 
55.593 

Total 
28 
2725.310 
Table 3: ANOVA table

Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
25.423 
3.7463 
6.7862 
0.0000 
17.7362 
33.1096 
X Variable 1 
0.2845 
0.0606 
4.6928 
0.0001 
0.1601 
0.4089 
Table 4: Summary table
Here, the researcher provides detailed discussion of the results founds from the analysis. Description of all the aspects was mentioned below in a sequence manner.
Referred to the table 1 as mentioned above in the result section, it can be said that the table provides brief of the descriptive statistics related to both the variables. From this table, the mean of retention rate is found as 57.41% and the mean of graduation rate is found as 41.76%. Since, the mean is the measure of central tendency; it can be concluded that all the selected 29 universities experienced more than 50 percent retention rate, which means over and above half of the student who got admission in the respective universities in several courses, are staying with the universities till end of their study.
Again, the results mentioned in the table 1 also indicates that the average graduation rate of these selected university remain below the 50 percent, which means each year, the number of students who graduated from their respective universities are lower than half of its student got admission in various curses (Kobayashi, Mark and Turin, 2012).
If the standard deviation is being considered as here, then it can be said that the standard deviation mainly explores the deviation of data from its average value (Montgomery, Peck and Vining, 2012). So, if the current context is taken into account, then the results shows in table 1 indicates that the retention rate of the selected universities experience more deviation in against the graduation rate of those universities. This indicates that retention rate fluctuated more frequently than graduation rate.
Method
Finally, the results shown in table 1 in terms of minimum and maximum statistics, it can be said that the spread of data related to retention rate is more than the spread of graduation rate. In case of retention rate, the minimum value is 4 %; whereas the maximum value of this variable is 100 %. Therefore, the spread of retention rate is 96 %. On the other hand, the maximum value of the graduation rate is 61 %; whereas the minimum value is 25 %. Therefore, the spread of this variable is 36 %. Hence, it can be said that the retention rate is fluctuated almost double to graduation rate. This also supports the conclusion regarding deviation of the results drawn from standard deviations.
Figure 1 mentioned in the above result section provides the scatter diagram of graduation rate and retention rate with retention rate being the independent variable. It is the fact that explanation of findings found from the scatter plot requires understanding of four important aspects such as direction, form, strength and outliers. Under such circumstances, the scatter plots as drawn in the above result section related to retention rate and graduation rate with retention rate being independent variable are discussed here in terms of these four aspects:
[a] Direction:
Mainly two types of directions are explained while discussing the direction of the scatter plot, such as positive direction and negative direction. Here, the positive direction indicates that larger values of the category variable will be associated with larger value of response variable. On the other hand, negative direction means larger values of the category variable will be associated with smaller value of response variable.
In this context, since the larger values of the category variable [retention rate] are associated with larger value of response variable [graduation rate]; the scatter plot indicates a positive direction.
[b] Form:
On the basis of form, any scatter diagrams are segmented as no association, no linear association, linear association and perfect linear association.
In this context, assessment of the scatter plots between retention rate and graduation rate, a linear association is found here.
[c] Strength:
According to strength, any scatter plot can be segmented as
 Zero correlation (r = 0 or near to zero);
 Weak correlation (0.35 < r < +0.35);
 Moderate correlation (0.35 < r < 0.55, 0.35 < r <0.55); and
 Strong relation (± 0.55 < r < ± 1)
Therefore, in this context, the scatter plots between retention rate and graduation rate shows a moderate association.
Finally, the scatter plot is explained here as the moderately positive linear correlation.
Here, the regression equation is as follow:
Graduation rate = Intercept coefficient + coefficient of retention rate * retention rate
Using the results shown in table 2, 3 and 4, the coefficient of intercept is 25.423 and coefficient of retention rate is 0.2845.
So, the estimated regression equation becomes
Graduation rate = 25.423 + 0.2845 * retention rate
The slope of a line is a dimension of how many units it goes up as well as down for each unit is change to the right. Slope of any line can be zero, positive or negative (Pardoe, 2012).
Here, if the slope is negative, it means the correlation between entry price and week is negative and positive means there is a positive relation between entry price and week.
Result
From the table 4, the value of the intercept is found as 25.423. It indicates that the expected mean value of graduation rate will be 25.423% if the retention rate is being changed to 0%.
Again, from table 3, it is noted that the value of coefficient of determination is 0.449. This is a precise parameter which provides details about the goodness of fit of the regression model.
It is noted that any value of coefficient of determination near 1 means the estimation is appropriate, whereas any value below 1 means estimation is not efficient enough to predict the graduation rate on the basis of retention rate.
Under such circumstances, it can be said that there is a 44.9% chance that the estimation will be accurate. However, the model is not a good fit.
Therefore, it is concluded that the provided data is not effective one to estimate the graduation rate with the help of retention rate. Therefore, more review is necessary to predict this specific case.
Under such circumstances, if the case scenario of the South University is being considered here, then it can be found that here the retention rate is 51 %, however, the graduation rate reach to 25 %. This would indicate that almost 50 percent of the student, the university is being able to retain are graduated in their specified subject. Since, the overall findings obtained from the study also indicates that same association, in this specific case, it can be said that being the president of South University, no concern is there regarding the performance of this university in against the other online universities (Sprinthall, 2012).
Right at the same time, if the circumstance of the University of Phoenix is being considered here, then it can be found that here the retention rate is 4 %, however, the graduation rate reach to 28 %. Though, this indicates the graduation rate is far better than the retention rate, in specific, there is no valid conclusion can be drawn from these two figures (XUE, 2010). In addition to this, it can also be concluded that such information is also largely deviated from the results obtained in the study. Hence, under such circumstances, it can be said that being the president of the University of Phoenix, a concern is there regarding the performance of this university in against the other online universities.
The study experienced a moderate positive correlation between retention rate and graduation rate, which further explores that the regression model cannot be considered as the best fit. So, there is an indication that graduation rate might influenced by other variables associated with the online education providing universities such as quality of education, facilities etc. Therefore, to reach into a meaningful conclusion regarding both the variable, other variables also needs to be take care of while establishing the multiple regression model.
References
Bekker, P. and Kleibergen, F. (2001). Finitesample instrumental variables inference using an asymptotically pivotal statistic. Amsterdam: Tinbergen Institute.
Gionis, A. (2013). Data Analysis. Data Science Journal, 12(0), pp.GRDI13GRDI18.
HaÃŒË†rdle, W. and Simar, L. (2012). Applied multivariate statistical analysis. Berlin: Springer.
Kamath, C. (2009). ApplicationDriven Data Analysis. Statistical Analysis Data Mining, 1(5), pp.285285.
Kim, G. and Chambers, R. (2011). Regression Analysis under Probabilistic MultiLinkage. Statistica Neerlandica, 66(1), pp.6479.
Kobayashi, H., Mark, B. and Turin, W. (2012). Probability, random processes, and statistical analysis. Cambridge: Cambridge University Press.
Montgomery, D., Peck, E. and Vining, G. (2012). Introduction to linear regression analysis. Hoboken, NJ: Wiley.
Pardoe, I. (2012). Applied regression modeling. Hoboken, NJ: Wiley.
Sprinthall, R. (2012). Basic statistical analysis. Boston: Pearson Allyn & Bacon.
XUE, L. (2010). Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data. Scandinavian Journal of Statistics, 37(4), pp.644663.
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