Question

The following is the regression output for a study on serious crime rates in cities in the United States. The data was collected in the 1970s. AREA is the size of the city, DOCS is the number of Doctors, % > 65 is the proportion of residents greater than 65 years of age and HOSPITAL BEDS is the number of hospital beds per population. Answer the following questions given the supplied regression output. These variables are regressed on the serious crime rates in an attempt to explain variance in crime rates in different cities. (3 points per question)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.673273
R Square	0.4532966
Adjusted R Square	0.4047007
Standard Error	12.033733
Observations	50
ANOVA
	df	SS	MS	F	Significance F
Regression	4	5403.1119	1350.778	9.3278852	1.40799E-05
Residual	45	6516.4834	144.81074
Total	49	11919.595
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	66.550797	9.4140949	7.069272	8.044E-09	47.58983709	85.511758
AREA	1.9887428	0.8666883	2.2946459	0.0264716	0.24314292	3.7343427
% > 65	-1.368696	0.6703271	-2.041833	0.0470559	-2.718804364	-0.018588
DOCS	8.6790607	3.3369629	2.6008862	0.1125379	1.958072413	15.400049
HOSP BEDS	-2.407649	0.7129517	-3.377016	0.0015198	-3.843607893	-0.971691

1. How accurate is regression model? What amount of the variance in Serious Crimes is explained
2. How strong is the linear relationship between the predictors and Serious Crimes?
3. How close to the regression line is the majority of observations?
4. Does the model fit the data better than not using these predictor variables? Is the F-test significant? What hypothesis can we draw from it?
5. Use the output to form the regression equation.
6. Are all the predictors significant? Is zero contained in the confidence intervals? Why is that interesting?
7. Of the significant variables, does the corresponding slopes make intuitive sense?   Are the effects of the predictors reasonable?
8. How much variance in serious crime is unaccounted for? What additional data might explain some of this remaining variance?

Answer 1

How accurate is regression model? What amount of the variance in Serious Crimes is explained

R^2 = 0.4533

hence 45.33 % of variance is explained by this model

How strong is the linear relationship between the predictors and Serious Crimes?

significance F = 1.4080*10^(-5) << 0.05

hence the model is significant

How close to the regression line is the majority of observations?

standard error = 12.0337

Does the model fit the data better than not using these predictor variables? Is the F-test significant? What hypothesis can we draw from it?

yes, as F-test significant

Are all the predictors significant? Is zero contained in the confidence intervals? Why is that interesting?

yes , all predictors are significant as p-value < 0.05 for all 4 variable

zero is not present in confidence intervals ,this is interesting because p-value < 0.05 in each case and 0 is not present in any CI

Of the significant variables, does the corresponding slopes make intuitive sense?   Are the effects of the predictors reasonable?

yes, slope make intuitive sense

How much variance in serious crime is unaccounted for? What additional data might explain some of this remaining variance?

unaccounted = 1 - 0.4533 = 0.5467