Question

Illinois State is worried about the rise in crimes in the state. The State Crime Control Unit engaged the services of a researcher to identify the factors that are contributing to rise in crime in the state. The researcher collected data for 50 cities within Illinois to ascertain the cause of rising crime. She collected data on seven variables including overall crime rate per 1 million residents in each of the 50 cities and reported violent crime rate per 100,000 residents.

The details of variables on which she collected data are as following.

Variable Name	Description
Crime Rate	Overall crime rate per 1 million residents
Violent Crime Rate	Reported violent crime rate per 100,000 residents
Funding	Annual police funding in $/resident
High School	% of people 25 years+ with 4 yrs. of high school
School Dropout	% of 16 to 19 year-olds not in high school and not high school graduates
Undergraduate	% of 18 to 24 year-olds in college
Graduate	% of people 25 years+ with at least 4 years of college

1. provide interpretation for following:

R²:

Adjusted R²:

Coefficient of Funding:

Coefficient of undergraduate:

Standard Error of Estimate:

Regression Analysis
Regression Statistics
Multiple R	0.5701
R Square	0.3250
Adjusted R Square	0.2810
Standard Error	249.2410
Observations	50
ANOVA
	df	SS	MS	F	Significance F
Regression	3	1376029.9687	458676.6562	7.3836	0.0004
Residual	46	2857569.9513	62121.0859
Total	49	4233599.9200
	Coefficients	Standard Error	t Stat	P-value	Lower 90%	Upper 90%
Intercept	612.8399	232.7230	2.6333	0.0115	222.1770	1003.5028
Funding	11.9794	2.7266	4.3935	0.0001	7.4024	16.5565
Undergraduate	0.3722	2.5706	0.1448	0.8855	-3.9429	4.6873
High School	-6.0944	3.6949	-1.6494	0.1059	-12.2969	0.1080

Answer 1

R²=0.3250 i.e. 32.50% of total variation in Crime Rate is explained by this regression equation.

Adjusted R²=0.2810 i.e. 28.1% of the variation in Crime Rate is explained by the model, adjusted for the number of predictors in the model relative to the number of observations. Note that R² always increases when we add a new predictor to the model, even when there is no real improvement to the model. However the value of adjusted R² increases only when the new term improves the model.Thus the adjusted R² value incorporates the number of predictors in the model to choose the correct model.

Coefficient of Funding=11.9794 i.e.  the response variable (i.e. Crime rate) is increased by 11.9794 (per 1 million residents) when Funding is increased by $1/resident while holding other predictors in the model constant.

Standard Error of Estimate is the average distance that the observed values fall from the regression line. Smaller values are better because it indicates that the observations are closer to the fitted line. Here it is 249.2410 which is too high hence fitting is not good.