Question

Develop estimated regression equations, first using annual income as the independent variable and then using household size as the independent variable. Which variable is the better predictor of annual credit card charges? Discuss your findings -

Income ($1000s)	Household Size	Amount Charged ($)
54	3	4,016
30	2	3,159
32	4	5,100
50	5	4,742
31	2	1,864
55	2	4,070
37	1	2,731
40	2	3,348
66	4	4,764
51	3	4,110
25	3	4,208
48	4	4,219
27	1	2,477
33	2	2,514
65	3	4,214
63	4	4,965
42	6	4,412
21	2	2,448
44	1	2,995
37	5	4,171
62	6	5,678
21	3	3,623
55	7	5,301
42	2	3,020
41	7	4,828
54	6	5,573
30	1	2,583
48	2	3,866
34	5	3,586
67	4	5,037
50	2	3,605
67	5	5,345
55	6	5,370
52	2	3,890
62	3	4,705
64	2	4,157
22	3	3,579
29	4	3,890
39	2	2,972
35	1	3,121
39	4	4,183
54	3	3,730
23	6	4,127
27	2	2,921
26	7	4,603
61	2	4,273
30	2	3,067
22	4	3,074
46	5	4,820
66	4	5,149

Answer 1

INCOME---------------------------------------------------

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	2174	198203	10374.48	42699148.8	419956.56
mean	43.48	3964.06	SSxx	SSyy	SSxy

sample size ,   n =   50          
here, x̅ = Σx / n=   43.48   ,     ȳ = Σy/n =   3964.06  
                  
SSxx =    Σ(x-x̅)² =    10374.4800          
SSxy=   Σ(x-x̅)(y-ȳ) =   419956.6          
                  
estimated slope , ß1 = SSxy/SSxx =   419956.6   /   10374.480   =   40.4798
                  
intercept,   ß0 = y̅-ß1* x̄ =   2203.9996          
                  
so, regression line is   Ŷ =   2203.9996   +   40.4798   *x
                  
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    25699404.034          
                  
std error ,Se =    √(SSE/(n-2)) =    731.713          
                  
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.6310          
                  
R² =    (Sxy)²/(Sx.Sy) =    0.3981   

HOUSEHOLD-----------------------------------------------------------------------

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	171	198203	148.18	42699148.8	59883.74
mean	3.42	3964.06	SSxx	SSyy	SSxy

sample size ,   n =   50          
here, x̅ = Σx / n=   3.42   ,     ȳ = Σy/n =   3964.06  
                  
SSxx =    Σ(x-x̅)² =    148.1800          
SSxy=   Σ(x-x̅)(y-ȳ) =   59883.7          
                  
estimated slope , ß1 = SSxy/SSxx =   59883.7   /   148.180   =   404.1284
                  
intercept,   ß0 = y̅-ß1* x̄ =   2581.9410          
                  
so, regression line is   Ŷ =   2581.9410   +   404.1284   *x
                  
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    18498431.339          
                  
std error ,Se =    √(SSE/(n-2)) =    620.793          
                  
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.7528          
                  
R² =    (Sxy)²/(Sx.Sy) =    0.5668   

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As you can see Household R sqyare is better and hence better predictor of annual credit card charges

Please revert back in case of any doubt.

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