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### Question

**Background**

One of the biggest challenges in the higher education sector has been the recent growth of online universities. The Online Education Database is an independent organisation whose mission is to build a comprehensive list of accredited online colleges. The **Excel spreadsheet** (OnlineEdu.xlsx) contains data on the** retention rate** (%) and the **graduation rate** (%) for **29 online colleges in the United States.**

**Instructions**

**Prepare a 2,000 word report** that addresses the following 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?

### Solution

All the following data calculations are done using the **Data Analysis tool** in MS Excel.

**Descriptive analysis of the two variables (retention and graduation rates in %)**– When the data relate to a variable, the summarization process extends to descriptive measures of statistics. Two important features are the central tendency (mean or average, median, mode) and dispersion (range, standard deviation, standard error).

Here, the mean or average gives the central value of a data set and range or standard deviation gives the degree of scattering in the values of a data set. This is done using the **Descriptive Statistics** function in the **Data Analysis tool.**

**Scatter diagram with retention rate as the independent variable**– A scatter plot or diagram is used for paired numerical data to determine whether there exists any linear relationship between the two variables.**Developing and estimating a regression equation to predict the graduation rate (%) given the retention rate (%)**– A regression equation allows one to predict the value of one variable (dependent, say y) given the value of another variable (independent, say x). Here, y and x are represented in a mathematical form like- y = a + bx. In this case, let y be the graduation rate and x be the retention rate.

The parameters a and b are determined from the given data such that, the sum of squares of the errors of estimation (of x and y) are the least. Here, a is called the intercept coefficient and b is the slope coefficient.

**Statistical significance in the relationship between the two variables**– Statistical significance is the prospect that a relationship exists not merely because of random chances but something more than that. The F and P values determine the statistical significance. An F value gives the ratio between two mean square values, where more the F value, more is the variation among group means. These values are obtained from the ANOVA table after applying the function Regression. A P value is the estimated**probability of rejecting**a null hypothesis when that hypothesis is true. The null hypothesis, in this case, is that there exists no statistical significance between the two variables. The alternative hypothesis says that there exists a significant relationship between these two variables. In this case, as the general rule, the significance level of both F and P are fixed at 0.05, for which any value less than this would result in rejection of the**null hypothesis**and acceptance of the alternative hypothesis (Walker & Lev 1958).

The regression line in this model is given by the equation: y = 25.42 + 0.28x (using approximated values), where 25.42 is the intercept coefficient and 0.28 is the slope coefficient. The regression equation implies that for every unit increase in the retention rate, the graduation rate (%) increases by 0.28 (approx.). The intercept is basically the expected mean value of the predicted variable when all the values of the predictor variable are zero. Interpretation of the intercept value says that if retention rate of students (x) takes the value zero, then the graduation rate would be 25.42%. Mathematically, however, correct this figure might be, practically it is not possible for the retention rate to take 0 values, as that would make any college collapse. Hence, in reality, this value has no inherent meaning.

**Statistical significance of the association between graduation rate (%) and retention rate (%)**Whether the association between the graduation rate and retention rate is statistically significant or not can be found by checking the Significance F value and P values given in the**ANOVA table**(Table**3**). It is known that the result values are reliable if the F value is less than 0.05 (for a 95% confidence interval). Here, the Significance F value and P-value is 0.0000695 (approx.). As this value is far below the critical value, hence the null hypothesis is rejected. Thus, there exists a statistically significant association between these two variables.**……….**

The goodness of fit of a regression line in a model describes how well the model fits a set of observations (Olivares & Forero 2010).

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