Question

Problem 3: A linear regression by using famous data set found in Freedman et al. (1991) in Table 1: ‘Statistics’ refers to the percapita consumption of cigarettes in various countries in 1930 and the death rates (number of deaths per million people) from lung cancer for 1950.

Table 1: Death rate data in in Freedman
Obs	Country	Cigarette	Deaths per million
1	Australia	480	180
2	Canada	500	150
3	Denmark	380	170
4	Finland	1100	350
5	GreatBritain	1100	460
6	Iceland	230	60
7	Netherlands	490	240
8	Norway	250	90
9	Sweden	300	110
10	Switzerland	510	250
11	USA	1300	200

1. Perform the simple linear regression with and without USA and make the overlay graph.
2. The question is: should we use USA data, should the regression line pass through the original? Give your answer.

Answer 1

Here, Number of cigarettes consumed is independent variable => x

Deaths per million is dependent variable => y

Thus, we have

	X	Y	X^2	Y^2	XY
	480	180	230400	32400	86400
	500	150	250000	22500	75000
	380	170	144400	28900	64600
	1100	350	1210000	122500	385000
	1100	460	1210000	211600	506000
	230	60	52900	3600	13800
	490	240	240100	57600	117600
	250	90	62500	8100	22500
	300	110	90000	12100	33000
	510	250	260100	62500	127500
	1300	200	1690000	40000	260000
Total	6640	2260	5440400	601800	1691400

We have, y = a + bx where

Thus,

a = (2260*5440400 - 6640*1691400)/ (11*5440400 - 6640^2) = 67.56

b = (11*1691400 - 6640*2260)/ (11*5440400 - 6640^2) = 0.228

Thus, y = 67.56 + 0.228*x

Now, this includes USA data.

For the regression line pass through the origin, its intercept should be 0. Here, it is 67.56 and not zero

Thus, regression line wont pass through the origin

Now, if we exclude USA data, we get

	X	Y	X^2	Y^2	XY
	480	180	230400	32400	86400
	500	150	250000	22500	75000
	380	170	144400	28900	64600
	1100	350	1210000	122500	385000
	1100	460	1210000	211600	506000
	230	60	52900	3600	13800
	490	240	240100	57600	117600
	250	90	62500	8100	22500
	300	110	90000	12100	33000
	510	250	260100	62500	127500
Total	5340	2060	3750400	561800	1431400

We have, y = a + bx where

Thus,

a = (2060*3750400 - 5340*1431400)/ (10*3750400 - 5340^2) = 9.14

b = (10*1431400 - 5340*2060)/ (10*3750400 - 5340^2) = 0.369

Thus, y = 9.14 + 0.369*x