Modern housing areas often seek to obtain, or at least convey, low rates of crimes. Homeowners and...

Question

Modern housing areas often seek to obtain, or at least convey, low rates of crimes. Homeowners and families like to live in safe areas and presumably are willing to pay a premium for the opportunity to have a safe home.

These data from Philadelphia Magazine summarize crime rates and housing prices in communities near and including Philadelphia. The housing prices for each community are the median selling prices for homes sold in the prior year. The crime rate variable measures the number of reported crimes per 100,000 people living in the community.

Exclude the data for Center City, Philadelphia from this analysis. It’s a predominantly commercial area that includes clusters of residential housing. The amount of commercial activity produces a very large crime rate relative to the number of residents.

The data sets for this problem can be found on theData Sets page for the text book(Links to an external site.). See Chapter 20; 4M Analytics: Crime and Housing Prices in Philadelphia (Data set available:
20_4m_philadelphia

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Question 1

A)How could local political and business leaders use an equation that relates crime rates to housing values to advocate higher expenditures for police?

Think about what might happen when more money is spent on police – both the direct effect and a guess about what might happen with property values. How can the data be used to advocate for higher expenditures?

What is the impact of higher property values on city operations?

B)How could the data be used for higher expenditures? What is the impact of higher property values on city operations?

C)Would an equation from these data produce a causal statement relating crime rates to housing prices? Explain.

HINTTake a look at the equation(s) in the attached plots for questions 5 and 6 and explain the relationship

question 2

A)For modeling the association between crime rates and housing prices, explain why a community leader should consider crime rates the explanatory variable.Think about what it means to be an explanatory variable.

B)Do you anticipate differences in the level of crime to be linearly related to differences in the housing prices? Explain in the context of your answer the underlying implication of a linear relationship.

C)Also explain in the context of your answer the underlying implication of a linear relationship. In other words, how does one interpret a linear relationship?
question 3

A)Examine the above scatterplot for the housing prices on crime rates. Describe the association. Is it strong? What is the direction?

B)Interpret the slope, intercept, and summary statistics (R-Squared and the Standard Error).

C)Do you think that there is a better fit with this model than with the regression ofHousing Price on Crime Rate. What is the natural interpretation of the reciprocal onCrime Rate?

question 4

A)Which model do you think offers the better summary of the association between crime rates and housing prices? Use residual plots, summary statistics, and substantive interpretation to make your case.

B)Choose the equation that you think best summarizes the relationship between crime rates and housing prices. Interpret the regression coefficients--the intercept and the slope for both the un-transformed and transformed models.

C)Does an increment in the crime rate from 1 to 2 per 100,000 have the same impact (on average) on housing prices as the change from 11 to 12 per 100,000?

Hint: Use the model equations to quantify the impact on housing prices for the given change in crime rate.

Pritam · Accepted Answer

Question 1:
A. A linear model could be suggested by the leaders to describe the association between the crime rate and the housing price. The further the house is from a crime hot spot, the more the prices get increased. 
B. The same concept can be explained by the data provided here. One can see that with the increase in the crime rate the housing prices get lowered and hence an inverse relationship could be estimated from the data as well. 
C. Though the linear association may not be sufficient for any causal statement but the association could be explained by the unit change in the corresponding variables. 
Question 2:
A. An explanatory variable is generally used to predict a response variable which may be correlated with the explanatory variable and thus the variance in the response variable could be explained by the explanatory variable. In this context one is interested in the housing prices and lots of factors could have a considerable impact on the pricing. People generally tend to have a safe ambiance and also the quality of life. Clearly, the crime rate directly or indirectly impacts the price of the household and hence the community leader should take the crime rate as a potential explanatory variable to model the association between the price of the housing and the crime rate.
B. To check the linear dependence one could use the scatterplot between the crime rate and the pricing of the house and the following graph is given below.
As it seems that the variables are not quite linearly related. Also, one data point seems to be far away from the other points and thus this could be considered as an outlier. Though there may be some other relationship between these two variables like polynomial or some other non-linear relationships. 
C. To determine the strength of a linear relationship between two variables, one can easily interpret the result. As the linear equation can be described in terms of the difference of the variables. One unit change in the explanatory variable is accompanied by the amount of the coefficient (of the response variable) in the response variable. To determine the relationship between two variables, one could think about the simplest linear model as that could be interpreted quite easily.
Question 3:
A. From the scatterplot, one can say that the linear relationship is negative in direction as expected intuitively. Also, a potential outlier could be seen in the scatterplot which is distant from all other data points. The weak negative relationship implies that some transformation of the explanatory variable is needed to capture the non-linear relationship between the variables. 
B. The slope and the intercept can be seen from the trendline in the scatterplot. The intercept of the linear model is given by 176629 which implies that with a zero crime rate, the average housing price is 176629 dollars nearly. The slope is found to be -576.91 which implies that with one unit increase in the crime rate, the average housing price will be decreased by almost 577 dollars. The R-squared is found to be 0.0625 which describes the fact that only 6% of the variance in the response variable can be explained by the explanatory variable which is not quite good and hence the model is not quite a good fit to the data. 
C. For the transformed explanatory variable or the inverse transformation on the crime rate, actually gives a better model than the previous one. Here the R-squared is found to be nearly 17% and hence a better amount of variance of the response variable is explained by the explanatory variable. The F-statistic of the overall model has a significant p-value at a 5% significance level and hence the overall model fit is also better compared to the previous one. Hence the second model seems to be the best.
Question 4:
A. The second model is the best one to use as far as summary statistics of both the models are concerned. The summary statistics for both the models are given below.
	SUMMARY OUTPUT for 1-st model
	
	
	
	
	
	
	
	
	
	
	
	Regression Statistics
	
	
	
	
	
	Multiple R
	0.249961
	
	
	
	
	
	R Square
	0.06248
	
	
	
	
	
	Adjusted R Square
	0.052815
	
	
	
	
	
	Standard Error
	84325.05
	
	
	
	
	
	Observations
	99
	
	
	
	
	
	
	
	
	
	
	
	
	ANOVA
	
	
	
	
	
	
	 
	df
	SS
	MS
	F
	Significance F
	
	Regression
	1
	45967293329
	45967293329
	6.464511
	0.01258726
	
	Residual
	97
	6.89739E+11
	7110714758
	
	
	
	Total
	98
	7.35707E+11
	 
	 
	 
	
	
	
	
	
	
	
	
	 
	Coefficients
	Standard Error
	t Stat
	P-value
	Lower 95%
	Upper 95%
	Intercept
	176629.4
	11245.58821
	15.7065513
	2.15E-28
	154310.0285
	198948.8
	Crime Rate
	-576.9081
	226.9022593
	-2.542540253
	0.012587
	-1027.246303
	-126.57
	SUMMARY OUTPUT for 2-nd model
	
	
	
	
	
	
	
	
	
	
	
	Regression Statistics
	
	
	
	
	
	Multiple R
	0.417028
	
	
	
	
	
	R Square
	0.173912
	
	
	
	
	
	Adjusted R Square
	0.165396
	
	
	
	
	
	Standard Error
	79155.23
	
	
	
	
	
	Observations
	99
	
	
	
	
	
	
	
	
	
	
	
	
	ANOVA
	
	
	
	
	
	
	 
	df
	SS
	MS
	F
	Significance F
	
	Regression
	1
	1.28E+11
	1.28E+11
	20.42091
	1.75E-05
	
	Residual
	97
	6.08E+11
	6.27E+09
	
	
	
	Total
	98
	7.36E+11
	 
	 
	 
	
	
	
	
	
	
	
	
	 
	Coefficients
	Standard Error
	t Stat
	P-value
	Lower 95%
	Upper 95%
	Intercept
	97920.6
	15462.18
	6.332909
	7.51E-09
	67232.44
	128608.8
	Rec crime rate
	1301376
	287982
	4.51895
	1.

Modern housing areas often seek to obtain, or at least convey, low rates of crimes. Homeowners and families like to live in safe areas and presumably are willing to pay a premium for the opportunity...

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