Question

From the data in the table below, estimate two regression models using a calculator or MS Excel (calculations only) to solve this. At a very minimum, please attempt one using a calculator. Show your work in a table Model 1: Weekly Disposable IncomeAge+ Model 2: Weekly Per Capita Consumption Age+ Age (years) Income (S Weekly Disposable Weekly Consumption per capita (S 30 35 40 45 50 500 550 600 500 900 1000 1000 1200 1500 50 60 60 50 70 75 80 90 60 65 70 For each model a) Find the slope, intercept and model equation. b) Interpret the estimated regression line c) Find and interpret R2 d) Find the predicted value of the dependent variable when Age is 58.

Answer 1

for weekly disposable income-

X(age)	Y(weekly disposable income)	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
30	500	400	130401.2	7222.222
35	550	225	96790.12	4666.667
40	600	100	68179.01	2611.111
45	500	25	130401.2	1805.556
50	900	0	1512.346	0
55	1000	25	19290.12	694.4444
60	1000	100	19290.12	1388.889
65	1200	225	114845.7	5083.333
70	1500	400	408179	12777.78

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	450	7750	1500	988888.9	36250
mean	50	861.111111	SSxx	SSyy	SSxy

sample size ,   n =   9          
here, x̅ =   50       ȳ =   861.1111111  
                  
SSxx =    Σ(x-x̅)² =    1500          
SSxy=   Σ(x-x̅)(y-ȳ) =   36250          
                  
slope ,    ß1 = SSxy/SSxx =   24.16666667          
                  
intercept,   ß0 = y̅-ß1* x̄ =   -347.2222222          
                  
so, regression line is   Ŷ =   -347.2222   +   24.1667   *x
                  

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.9412          
                  
R² =    (Sxy)²/(Sx.Sy) =    0.8859  

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a)

slope = 24.167

intercept= -347.2222

eqn is

Ŷ =   -347.2222   +   24.1667   *x

b) for every unit increase in age, weekly disposable income will get increase by 24.1667

c) R² = 0.8859

88.59% variations in observations of variable Y(weekly disposable income) is explained by variable X(age)

d)

age,X=58

Ŷ =   -347.2222   +   24.1667   *58=1054.444

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for weekly consumption per capita-

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
30	50	400	241.9753	311.1111
35	55	225	111.4198	158.3333
40	60	100	30.8642	55.55556
45	60	25	30.8642	27.77778
50	50	0	241.9753	0
55	70	25	19.75309	22.22222
60	75	100	89.19753	94.44444
65	80	225	208.642	216.6667
70	90	400	597.5309	488.8889

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	450	590	1500	1572.222	1375
mean	50	65.5555556	SSxx	SSyy	SSxy

sample size ,   n =   9          
here, x̅ =   50       ȳ =   65.55555556  
                  
SSxx =    Σ(x-x̅)² =    1500          
SSxy=   Σ(x-x̅)(y-ȳ) =   1375          
                  
slope ,    ß1 = SSxy/SSxx =   0.916666667          
                  
intercept,   ß0 = y̅-ß1* x̄ =   19.72222222          
                  
so, regression line is   Ŷ =   19.7222   +   0.9167   *x
                     
                  
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.8954          
                  
R² =    (Sxy)²/(Sx.Sy) =    0.8017  

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a)

slope = 0.9167

intercept= 19.7222

eqn is

Ŷ =   19.7222   +   0.9167   *x

b) for every unit increase in age, weekly per capita consumption will get increase by 0.9167

c) R² = 0.8017

80.17% variations in observations of variable Y is explained by variable X(age)

d)

age,X=58

Ŷ =   19.7222   +   0.9167   *58 = 72.889