Graduation rates 2013 A prestigious college is interested in...

Graduation rates 2013 A prestigious college is interested in factors that might be associated with better graduation rates. The administrators wonder whether there is a relationship between acceptance rates and graduation rates. Before proceeding with their regression inference, what conditions and assumptions must be satisfied? Are they? (Source: US News and World Report National University Rankings October $2013 .$ http://colleges .usnews.rankingsandreviews.com/best-colleges/rankings /national-universities/data/spp' $\mathrm{p} \% 2 \mathrm{~B} 50 / \mathrm{page}+1)$ Response variable is Grad\% R-squared $=75.2 \%$ with $50-2=48$ $s=0.0254$ degrees of freedom $\begin{array}{lclll}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 0.975175 & 0.0074 & 131 & \leq 0.0001 \\ \text { Acceptance \% } & -0.284895 & 0.0236 & -12.1 & \leq 0.0001\end{array}$

Graduation rates 2013 A prestigious college is interested in factors that might be associated with better graduation rates. The administrators wonder whether there is a relationship between acceptance rates and graduation rates. Before proceeding with their regression inference, what conditions and assumptions must be satisfied? Are they? (Source: US News and World Report National University Rankings October $2013 .$ http://colleges .usnews.rankingsandreviews.com/best-colleges/rankings /national-universities/data/spp' $\mathrm{p} \% 2 \mathrm{~B} 50 / \mathrm{page}+1)$
Response variable is Grad\% R-squared $=75.2 \%$ with $50-2=48$ $s=0.0254$
degrees of freedom
$\begin{array}{lclll}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 0.975175 & 0.0074 & 131 & \leq 0.0001 \\ \text { Acceptance \% } & -0.284895 & 0.0236 & -12.1 & \leq 0.0001\end{array}$

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Key Concepts

Linearity

In a linear regression model, it is assumed that there is a linear relationship between the predictor (acceptance rate) and the response (graduation rate). This means that changes in the predictor are associated with proportional changes in the response variable.

Independence of Errors

This condition requires that the residuals (the differences between observed and predicted values) are independent of each other. It implies that there is no underlying structure, such as time-related patterns or clustering, that might cause the errors to correlate.

Homoscedasticity

Also known as constant variance, this assumption states that the variability of the residuals should be roughly the same across all levels of the predictor variable. It ensures that the model's predictions are equally reliable for all values of the predictor.

Normality of Errors

For accurate hypothesis testing and confidence intervals, it is assumed that the residuals are normally distributed. This assumption allows for the use of t-tests and F-tests in evaluating the model's coefficients.

Detection of Outliers and Influential Points

It is important to check that there are no outliers or influential observations that unduly affect the model’s fit. Such points can skew the analysis and may require further investigation or alternative modeling approaches.

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