Question

(please provide detailed solution with formula and figures)

the researcher considers using regression analysis to establish a linear relationship between the two variables – hours worked per week and yearly income. a) What is the dependent variable and independent variable for this analysis? Why? b) Use an appropriate plot to investigate the relationship between the two variables. Display the plot. On the same plot, fit a linear trend line including the equation and the coefficient of determination R2 . c) Estimate a simple linear regression model and present the estimated linear equation. Display the regression summary table and interpret the intercept and slope coefficient estimates of the linear model. d) Display and interpret the value of the coefficient of determination, R-squared (R2 ).

Hours Per Week	Yearly Income ('000's)
18	43.8
13	44.5
18	44.8
25.5	46.0
11.5	41.2
18	43.3
16	43.6
27	46.2
27.5	46.8
30.5	48.2
24.5	49.3
32.5	53.8
25	53.9
23.5	54.2
30.5	50.5
27.5	51.2
28	51.5
26	52.6
25.5	52.8
26.5	52.9
33	49.5
15	49.8
27.5	50.3
36	54.3
27	55.1
34.5	55.3
39	61.7
37	62.3
31.5	63.4
37	63.7
24.5	55.5
28	55.6
19	55.7
38.5	58.2
37.5	58.3
18.5	58.4
32	59.2
35	59.3
36	59.4
39	60.5
24.5	56.7
26	57.8
38	63.8
44.5	64.2
34.5	55.8
34.5	56.2
40	64.3
41.5	64.5
34.5	64.7
42.3	66.1
34.5	72.3
28	73.2
38	74.2
31.5	68.5
36	69.7
37.5	71.2
22	66.3
33.5	66.5
37	66.7
43.5	74.8
20	62.0
35	57.3
24	55.3
20	56.1
41	61.5

Answer 1

a)

Hours Per Week : it is independent variable as it does not depend on other

Yearly Income ('000's) : it dependent variable as it depend on other

.................

b

)R² =    (Sxy)²/(Sx.Sy) =    0.4435

........

c)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	1941.8	3726.3	4142.505538	4374.0	2834.77
mean	29.87	57.33	SSxx	SSyy	SSxy

sample size ,   n =   65          
here, x̅ = Σx / n=   29.87   ,     ȳ = Σy/n =   57.33  
                  
SSxx =    Σ(x-x̅)² =    4142.5055          
SSxy=   Σ(x-x̅)(y-ȳ) =   2834.8          
                  
estimated slope , ß1 = SSxy/SSxx =   2834.8   /   4142.506   =   0.6843
                  
intercept,   ß0 = y̅-ß1* x̄ =   36.8847          
                  
so, regression line is   Ŷ =   36.8847   +   0.6843   *x

.................

Regression Statistics
Multiple R	0.6660
R Square	0.4435
Adjusted R Square	0.4347
Standard Error	6.2159
Observations	65

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	36.8847	2.9863	12.3511	0.0000	30.9169	42.8524
X	0.6843	0.0966	7.0857	0.0000	0.4913	0.8773

intercept : if hours per week is 0, then yearly income will be 36.8847 units

slope: if x is increased by 1 unit , then income will increase by 0.6843 units

..........

R² =    (Sxy)²/(Sx.Sy) =    0.4435

44.35% of variation is explained by hours worked per week of yearly income

...................

THANKS

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