Linear Regression

Economics and Quantitative Analysis Linear Regression Report


The key purpose of this research report is to identify the relationship between the independent variable such as “Retention Rate” and the dependent variable such as ‘Graduation Rate”.

In addition, this report contains the simple linear regression analysis by which a connection between variables related to universities which provides the services online compared to other universities.


While discussing the universities, it is examined about the retention rate that the universities always try to retain their students in their institution through operating educational programs. In a similar manner, as per the graduation rate, it measures the percentage of students who are in the first year under the universities (Silverman, 2018).

As per this rate, it is found that these students complete their studies within the 150% of the provided time for the program. At this time, the economist and analyst will be attracted by the particular issue that is connected with the association between retention rate and the graduation rate.

In this way, it is essential to understand whether the first year student’s retention create an effect on accomplishing the educational program on time or not.

On the basis of requirements, it is necessary for the analyst to suggest or adopt effective educational policies and approaches in concern of online universities in order to develop the quality within the educational programs so that more students can be retained in online institutions.


In this report, the information of retention rate and the graduation rate of 29 online institutions in the US has been gathered and at the same time, this information has also been analyzed in the manner of examining the association between these two rates.In addition,

it is found that this collected data is completely quantitative and that is the reason, it needs to be measured statistically (Carroll, 2017).

In this way, the linear regression analysis can be adopted for these two rates such as retention rate and graduation rate of the universities as it helps the analyst to examine the relationship between independent and dependent variables through adopting linear regression equation.

As per this collected data and analysis objective, the regression equation is suitable for examining the relationship between two variables. In this, the regression equation can be in a different manner that is defined below:

Y = a + bX,

Y = Dependent Variable, while

X = Dependent variable.


Provide a descriptive analysis of the two variables

Descriptive StatisticsRR (%) GR (%) 
Standard Deviation23.249.87


On behalf of above table, it is examined that the descriptive statistic is done on the basis of data gathered from the 29 colleges of the USA.  In this way, the retention rate is 57.41% whereas the graduation rate is 41.76%. In addition, it is also determined that the standard deviation of retention rate is high in comparison to the graduation rate.

This defines that the data related to the retention rate has more deviation (Pyrczak, 2016). In this, the maximum and minimum retention rate is 100% and 40% while the maximum and minimum graduation rate is 60% and 25%.

Develop a scatter diagram with retention rate as the independent variable and about the relationship between the two variables

Above drawn diagram reflects the relationship between the retention rate and graduation rate as it represents a straight line that is from origin out to high x-y values that show a positive relationship between variables that means increased retention rate will lead the increased graduation rate (Hox et al., 2017).

Estimate a regression equation that can be used to predict the graduation rate (%) given the retention rate (%)

In this analysis, for predicting the graduation rate with the help of retention rate, a simple regression equation can be suitable that is given below:

Y = mX + C

Graduation Rate (%)M*Retention Rate (%) + C


State the estimated regression equation and interpret the meaning of the slope coefficient

Regression Statistics
Multiple R0.6702
R Square0.4492
Adjusted R Square0.4288
Standard Error7.4561
 dfSSMSFSignificance F
 CoefficientsStandard Errort StatP-valueLower 95%Upper 95%


Above table represents the regression on the basis of regression equation that is defined below:

Y = 0.2845x + 25.42
Graduation Rate (%) = 0.2845 * Retention Rate (%) + 25.42


Above calculation reflects that the graduation rate will be increased by 0.2845% while increasing 1% in retention rate (Shah et al., 2017). In this, the rate of change in Y because of change in X is examined by slope coefficient within the estimated regression equation.  In addition, the slope coefficient is identified as 0.666.

A statistically significant association between graduation rate (%) and retention rate (%)

The Hypothesis analysis is defined below:

H0: It means there is no relationship between the graduation rate and retention rate.

H1: Itmeans, there is a relationship between graduation rate and retention rate.


On behalf of above, the value of P is zero which is less as compared to significant value such as 0.05. It also defines that the null hypothesis is rejected in case of a relationship between two variables.

The regression equation provides a good fit

After analyzing the scatter diagram, the R square is 0.49% of the variation as per graduation rate that has a relationship with retention rate. In this way, the regression equation is considered as the good fit as the graduation rate is effectively predicted by retention rate (Giavarina, 2015).

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?

In this, retention rate and graduation rate of South Universities is identified as 51% and 25% respectively whereas the average retention rate and the graduation rate is calculated as 57.41% and 41.76% respectively in concern of all the universities (Kaytez et al., 2015).

It means the performance of the South Universities is low in against of other online universities because of the low retention rate that leans the low graduation rate.

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?

After accomplishing the analysis, it is found that the retention rate and graduation rate of Phoenix University is 4 % and 28 % while the average retention rate and the average graduation rate is determined as 57.41% and 41.76% in respect to all universities.

In this manner, there is need to keep the focus on University of Phoenix as compared to other online institutions because of low retention rate that leads the lower graduation rate (Hox et al., 2017).


On the basis of the above examination, it is discussed that there is found a significant relationship between retention rate (independent variable) and graduation rate (the dependent variable).

This discussion defines that there is a positive impact of the retention rate on the graduation rate. In addition, it is also depicted that students those are retained in the universities, complete their education program more efficiently (Yarnold and Linden, 2016).

At the same time, it is also determined in the analysis that South universities and the University of Phoenix have no effective performance in comparison to other universities.


After accomplishing the discussion above, it can be recommended to the universities those are not effectively performing their operation that some important changes related to educational approaches should be adopted by them in order to improve their educational programs as well as get effective results.

For this,  they need to use mix of more areas in their study material and incorporate new learning and teaching methods.


Carroll, R.J., 2017. Transformation and weighting in regression. UK: Routledge.

Giavarina, D., 2015. Understanding bland altman analysis. Biochemia medica: Biochemia medica25(2), pp.141-151.

Hox, J.J., Moerbeek, M. and Van de Schoot, R., 2017. Multilevel analysis: Techniques and applications. UK: Routledge.

Kaytez, F., Taplamacioglu, M.C., Cam, E. and Hardalac, F., 2015. Forecasting electricity consumption: A comparison of regression analysis, neural networks and least squares support vector machines. International Journal of Electrical Power & Energy Systems67, pp.431-438.

Pyrczak, F., 2016. Making sense of statistics: A conceptual overview. UK: Routledge.

Shah, S.A.A., Nadeem, U., Bennamoun, M., Sohel, F.A. and Togneri, R., 2017, January. Efficient Image Set Classification using Linear Regression based Image Reconstruction. In CVPR Workshops (pp. 601-610).

Silverman, B.W., 2018. Density estimation for statistics and data analysis. UK: Routledge.

Yarnold, P.R. and Linden, A., 2016. Novometric analysis with ordered class variables: The optimal alternative to linear regression analysis. Age7(64.05), pp.15-64.

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