Question

The table below contains real data for the first two decades of AIDS reporting.

Year	# AIDS cases diagnosed	# AIDS deaths
Pre–1981	91	29
1981	319	121
1982	1,170	453
1983	3,076	1,482
1984	6,240	3,466
1985	11,776	6,878
1986	19,032	11,987
1987	28,564	16,162
1988	35,447	20,868
1989	42,674	27,591
1990	48,634	31,335
1991	59,660	36,560
1992	78,530	41,055
1993	78,834	44,730
1994	71,874	49,095
1995	68,505	49,456
1996	59,347	38,510
1997	47,149	20,736
1998	38,393	19,005
1999	25,174	18,454
2000	25,522	17,347
2001	25,643	17,402
2002	26,464	16,371
Total	802,118	489,093

Graph "year" versus "# AIDS cases diagnosed" (plot the scatter plot). Do not include pre-1981 data. Perform linear regression. Write the equations. (Round your answers to the nearest whole number. Round r to four decimal places.)

(a)    Linear Equation: ŷ =  +  x

(b)    a =

(c)    b =

(d)    r =

(e)    n =

Answer 1

a.

X - Mx	Y - My	(X - Mx)2	(X - Mx)(Y - My)
-10.5	-36136.7727	110.25	379436.1136
-9.5	-35285.7727	90.25	335214.8409
-8.5	-33379.7727	72.25	283728.0682
-7.5	-30215.7727	56.25	226618.2955
-6.5	-24679.7727	42.25	160418.5227
-5.5	-17423.7727	30.25	95830.75
-4.5	-7891.7727	20.25	35512.9773
-3.5	-1008.7727	12.25	3530.7045
-2.5	6218.2273	6.25	-15545.5682
-1.5	12178.2273	2.25	-18267.3409
-0.5	23204.2273	0.25	-11602.1136
0.5	42074.2273	0.25	21037.1136
1.5	42378.2273	2.25	63567.3409
2.5	35418.2273	6.25	88545.5682
3.5	32049.2273	12.25	112172.2955
4.5	22891.2273	20.25	103010.5227
5.5	10693.2273	30.25	58812.75
6.5	1937.2273	42.25	12591.9773
7.5	-11281.7727	56.25	-84613.2955
8.5	-10933.7727	72.25	-92937.0682
9.5	-10812.7727	90.25	-102721.3409
10.5	-9991.7727	110.25	-104913.6136
		SS: 885.5	SP: 1549427.5

Sum of X = 43813
Sum of Y = 802027
Mean X = 1991.5
Mean Y = 36455.7727
Sum of squares (SS_X) = 885.5
Sum of products (SP) = 1549427.5

Regression Equation = ŷ = bX + a

c. b = SP/SS_X = 1549427.5/885.5 = 1750

b. a = M_Y - bM_X = 36455.77 - (1749.78*1991.5) = -3448225

a. ŷ = 1750X - 3448225

d.

X - Mx	Y - My	(X - Mx)2	(Y - My)2	(X - Mx)(Y - My)
-10.5	-36136.7727	110.25	1305866343	379436.1136
-9.5	-35285.7727	90.25	1245085757	335214.8409
-8.5	-33379.7727	72.25	1114209227	283728.0682
-7.5	-30215.7727	56.25	912992921.5	226618.2955
-6.5	-24679.7727	42.25	609091181.9	160418.5227
-5.5	-17423.7727	30.25	303587856.1	95830.75
-4.5	-7891.7727	20.25	62280076.78	35512.9773
-3.5	-1008.7727	12.25	1017622.415	3530.7045
-2.5	6218.2273	6.25	38666350.42	-15545.5682
-1.5	12178.2273	2.25	148309219.5	-18267.3409
-0.5	23204.2273	0.25	538436163.3	-11602.1136
0.5	42074.2273	0.25	1770240601	21037.1136
1.5	42378.2273	2.25	1795914147	63567.3409
2.5	35418.2273	6.25	1254450823	88545.5682
3.5	32049.2273	12.25	1027152969	112172.2955
4.5	22891.2273	20.25	524008286.1	103010.5227
5.5	10693.2273	30.25	114345109.5	58812.75
6.5	1937.2273	42.25	3752849.506	12591.9773
7.5	-11281.7727	56.25	127278395.9	-84613.2955
8.5	-10933.7727	72.25	119547386.1	-92937.0682
9.5	-10812.7727	90.25	116916054.1	-102721.3409
10.5	-9991.7727	110.25	99835522.23	-104913.6136
		SS: 885.5	Sum: 13232984861.864	SP: 1549427.5

X Values
∑ = 43813
Mean = 1991.5
∑(X - M_x)² = SS_x = 885.5

Y Values
∑ = 802027
Mean = 36455.773
∑(Y - M_y)² = SS_y = 13232984861.864

X and Y Combined
N = 22
∑(X - M_x)(Y - M_y) = 1549427.5

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 1549427.5 / √((885.5)(13232984861.864)) = 0.4526

e. n=22