Problem 11_20, pg.590

The manager of a certain commuter rail system wants to determine which factors have a significant impact on the demand for rides in the large city served by the transportation network. The variables that he wants to relate to the number of weekly riders on the city’s rail are price per ride, population of the city, disposable income per capita of the citizens and the parking rate in the city parking lots.

As I look at each variable, some of them are expected to be positively related to the number of riders per week by their nature. Others are in no way, or seem not to be, related to weekly rider-ship. First, examining price per ride, compared to the number of weekly riders, one would expect this to be a positive linear relationship, and therefore expect the signs of the coefficient to be positive. The second variable, population of the city, is one I would expect to have a negative relationship in one sense, and a positive relationship in another. The expected sign of the coefficient at first glance of the data seems to be negative. At first, in 1966, over half of the population is riding the rail system on a weekly basis, and as the years go by, just a little less, and then a little more than half the population is riding the commuter rail. This change could be more attributable to the increased popularity and affordability of cars than being directly related to the number of weekly riders. Next, the data on income suggest a positive linear relationship between the variables. As disposable income per capita rises, the number of weekly rail riders decreases, again implying that possibly with the introduction of affordable cars or other forms of transportation, less people rode the city rail system. The last variable is weekly rider-ship compared to the parking rate of city parking lots. As the price of parking goes up, fewer citizens are riding the rail (not related to number of weekly riders), which again can be attributed to other more affordable modes of transportation, and an expected positive sign of the coefficient.

Results of multiple regression for Weekly_Riders
Summary measures
		Multiple R		0.9776
		R-Square		0.9557
		Adj R-Square		0.9477
		StErr of Est		21.4867
ANOVA Table
		Source		df		SS		MS		F		p-value
		Explained		4		219260.4797		54815.1199		118.7301		0.0000
		Unexplained		22		10156.9277		461.6785
Regression coefficients
				Coefficient		Std Err		t-value		p-value		Lower limit		Upper limit
		Constant		124.4269		516.7803		0.2408		0.8120		-947.3109		1196.1648
		Price_per_Ride		-166.9641		52.0106		-3.2102		0.0040		-274.8275		-59.1006
		Population		0.6210		0.2751		2.2570		0.0343		0.0504		1.1915
		Income		-0.0472		0.0129		-3.6572		0.0014		-0.0740		-0.0204
		Parking_Rate		194.6798		36.6143		5.3170		0.0000		118.7463		270.6133
Results of multiple regression for Weekly_Riders
Summary measures
	Multiple R		0.9724
	R-Square		0.9455
	Adj R-Square		0.9384
	StErr of Est		23.3208
ANOVA Table
	Source		df		SS		MS		F		p-value
	Explained		3		216908.6379		72302.8793		132.9440		0.0000
	Unexplained		23		12508.7695		543.8595
Regression coefficients
			Coefficient		Std Err		t-value		p-value		Lower limit		Upper limit
	Constant		1289.6821		24.6300		52.3622		0.0000		1238.7311		1340.6331
	Price_per_Ride		-203.7555		53.6060		-3.8010		0.0009		-314.6477		-92.8632
	Income		-0.0691		0.0093		-7.4576		0.0000		-0.0882		-0.0499
	Parking_Rate		212.0409		38.8528		5.4575		0.0000		131.6679		292.4140

For first regression table, #Weekly Riders= 124.43-166.96Price per Ride+0.621Population- 0.047Income+ 194.68Parking Rate

For second regression table, #Weekly Riders=1289.68- 203.76Price per Ride- 0.069Income+ 212.04Parking Rate

Table of correlations
		Weekly_Riders	Price_per_Ride	Population	Income	Parking_Rate
	Weekly_Riders	1.000
	Price_per_Ride	-0.896	1.000
	Population	0.946	-0.936	1.000
	Income	-0.934	0.944	-0.971	1.000
	Parking_Rate	-0.825	0.955	-0.919	0.949	1.000

After getting the regression coefficients and the correlations, I graphed the relationships to see how they presented themselves.

After examining the variables in graph form, I found that the coefficients did not resemble what I thought the data suggested. I did multiple regression analyses with and without the population variable. At first, including the population variable, population and parking rate are the only positive coefficient variables. The other two, price per ride and income have negative coefficients. When I excluded the population variable (because I thought this was the variable that would have no relationship with the dependent variable number of weekly riders), I found that price per ride and income still have negative coefficients. But the P-values of these coefficients suggest that they should still be included in the analysis because they are all less than .05.

The correlations of the coefficients for price per ride, population, income, and parking rate are -0.896, 0.946, -0.934, and -0.825 respectively. These all suggest that there are strong relationships between the dependent and each independent variable.

Although this data does not show at all what was expected from the variables, only 5% and 6% respectively of the total variation in the number of weekly riders is not explained by the estimated multiple regression model.