# Linear Regression Analysis

Purpose

The purpose of this report is to determine the relationship between the ‘retention rate’ (the independent variable) and the ‘graduation rate’ (the dependent variable).

For this, a simple linear regression analysis method is used to determine the association between these variables and concerns related to performance of some online universities as compared to other universities.

Background

Retention rate shows the measurement of the rate at which the students retain in their educational program at the institution for the next years. At the same time, graduation rate measures the percentage of a school’s first-time, first-year undergraduate students who complete their educational program or degree within 150% of the given time for the program (Bawa, 2016).

The economists would be interested in this particular issue related to association between retention rate and graduation rate because it is quite important to know whether the retention of the first year students has an impact on the completion of their graduation or not on time (Crawford, 2015).

It can be significant for the economists to suggest or apply better educational policies and approaches for the online universities to make the education better and retain the students in their educational program at the online institution and impact their graduation rate (Schwerdt et al., 2017).

Method

In order to determine the association between the retention rate and graduation rate, the data related to retention rate and graduation rate of 29 online colleges in the United States has been collected and analyzed.

This is quantitative and measurable data that can be analyzed statistically through linear regression analysis (Quirk, 2018). This data analysis method is used to determine the relationship between dependent and independent variables by using regression equation.

In this method, a linear equation is fitted to observed data to determine relationship between two variables.  Regression equation can be presented in the form of Y = a + bX, here Y is dependent variable while X is independent variable.

Results

a) Descriptive analysis of the two variables

 Descriptive Statistics RR (%) GR (%) Mean 57.41 41.76 Standard Deviation 23.24 9.87 Minimum 4 25 Maximum 100 61 Count 29 29

From the above descriptive statistics, it can be analyzed that the data is collected from 29 colleges of the USA. At the same time, the average retention rate and the average graduation rate is 57.41% and 41.76% respectively.

At the same time, standard deviation is quite higher for the retention rate as compared to graduation rate means there are more deviation in the data set of retention rate from the mean value (Quinlan et al., 2019).

In addition, the maximum retention rate and the minimum retention rate is 100% and 4% respectively while the maximum graduation rate and the minimum graduation rate is 61% and 25%.

b) Scatter diagram with retention rate as the independent variable and the relationship between the two variables

The above scatter diagram indicates about the relationship between the retention rate and graduation rate. There is a straight line going from the origin out to high x- and y-values showing a linear and positive relationship between both variables. It means the increase in retention rate will also increase the graduation rate (Silverman, 2018).

c) A regression equation to predict the graduation rate (%) given the retention rate (%)

For this, a simple regression linear regression equation can be sued to predict the graduation rates through retention rate. The linear equation can be given as below:

Y= mX+C

Graduation rate (%)= m * Retention rate (%)+C (Quirk, 2018).

Here,

m is slope.

C is constant.

d) The estimated regression equation and interpret the meaning of the slope coefficient

Regression analysis:

 SUMMARY OUTPUT Regression Statistics Multiple R 0.6702 R Square 0.4492 Adjusted R Square 0.4288 Standard Error 7.4561 Observations 29 ANOVA df SS MS F Significance F Regression 1 1224.29 1224.29 22.02 7E-05 Residual 27 1501.02 55.59 Total 28 2725.31034 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 25.4229 3.7463 6.7862 0.0000 17.7362 33.1096 RR(%) 0.2845 0.0606 4.6928 0.0001 0.1601 0.4089

From the above regression table, it can be determined that the regression equation will be as follows:

Y=0.2845x+25.42

Graduation rate (%)= 0.2845 * Retention rate (%)+25.42

From the regression equation, it can be determined that one percentage increase in retention rate will increase 0.2845% in graduation rate. Here, the slope coefficient represents the rate of change in Y due to change in X (Quinlan et al., 2019).

In the estimated regression equation, m or slope coefficient is 0.2845 indicating the change in graduation rate by 0.2845 if the retention rate changes by 1%.

e) Is there a statistically significant association between graduation rate (%) and retention rate (%).

Hypothesis:

H0: there is no statistically significant association between graduation rate (%) and retention rate (%).

H1: there is a statistically significant association between graduation rate (%) and retention rate (%).

The above regression test shows that p-value is zero that is less than the significance value of 0.05 (at 95% level). It means the null hypothesis can be rejected as there is a statistically significant association between graduation rate (%) and retention rate (%).

f) Did the regression equation provide a good fit? Explain

From the scatter diagram, it can be determine that R square or coefficient of determination equal to 0.4492 indicating about 45% of the variation in graduation rate (the dependent variable) can be explained by the relationship to retention rate (the independent variable).

This would be considered a good fit to the data as retention rate will substantially predict graduation rate effectively (Mendenhall et al., 2016).

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?

After reviewing the results, it can be determined that the retention rate and graduation rate of South University is 51% and 25% respectively. But at the same time, the average retention rate and the average graduation rate is 57.41% and 41.76% respectively for all universities.

So, it can be interpreted that retention rate and graduation rate is low for the South University as compared to other universities. However, retention rate has an impact on graduation rate but there may be other factors influencing the graduation rate.

It means there are concerns about the performance of South University compared to other online universities due to having low retention rate of the students along with other factors leading to lower graduation rate.

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?

Based on results, it can be determined that the retention rate and graduation rate of the University of Phoenix is 4% and 28% respectively. But at the same time, the average retention rate and the average graduation rate is 57.41% and 41.76% respectively for all universities.

Therefore, it can be determined that retention rate and graduation rate is quite low for the University of Phoenix in comparison of other universities. But its performance is satisfactory as retention rate is low as compared to graduation rate. It means there are few concerns about the performance of the University of Phoenix compared to other online universities.

Discussion

Based on the above analysis, it can be discussed that there is a significant relationship between retention rate and graduation rate. It means there is a positive impact of the retention rate on the graduation rate. But model is not fully fitted as there are other factors influencing the graduation rate (Bawa, 2016).

Apart from this, some of the universities like South University and University of Phoenix are not performing better as compared to other universities as they need to consider some changes in their educational approaches to bring better results.

Recommendations

Based on the above discussion, it can be recommended that there is need to improve the quality of education by introducing new online learning and teaching methods.

At the same time, budget, cost of education and subject domain and needs of the students should be considered to improve the quality of education and enhance graduation rates.

References

Bawa, P. (2016). Retention in online courses: Exploring issues and solutions—A literature review. Sage Open6(1), 2158244015621777.

Crawford, G. A. (2015). The academic library and student retention and graduation: An exploratory study. portal: Libraries and the Academy15(1), 41-57.

Mendenhall, W. M., Sincich, T. L., & Boudreau, N. S. (2016). Statistics for Engineering and the Sciences, Student Solutions Manual. Chapman and Hall/CRC.

Quinlan, C., Babin, B., Carr, J., & Griffin, M. (2019). Business research methods. South Western Cengage.

Quirk, T. J. (2018). Excel 2016 in Applied Statistics for High School Students. Springer.

Schwerdt, G., West, M. R., & Winters, M. A. (2017). The effects of test-based retention on student outcomes over time: Regression discontinuity evidence from Florida. Journal of Public Economics152, 154-169.

Silverman, B. W. (2018). Density estimation for statistics and data analysis. Routledge.