Question

The data shown below for the dependent variable, y, and the independent variable, x, have been collected using simple random sampling. x 10 15 17 11 19 18 17 15 17 18 y 120 150 170 120 170 180 160 140 180 190 a. Develop a simple linear regression equation for these data. b. Calculate the sum of squared residuals, the total sum of squares, and the coefficient of determination. c. Calculate the standard error of the estimate. d. Calculate the standard error for the regression slope. e. Conduct the hypothesis test to determine whether the regression slope coefficient is equal to 0. Test using alphaαequals=0.10

Answer 1

Answer:

Given that,

The data shown below for the dependent variable, y, and the independent variable, x, have been collected using simple random sampling.

x: 10 15 17 11 19 18 17 15 17 18

y: 120 150 170 120 170 180 160 140 180 190

(a).

Develop a simple linear regression equation for these data:

x	y	(x-)	(y-)	(x-)2	(y-)2	(x-)(y-)
10	120	-5.7	-38	32.49	1444	216.6
15	150	-0.7	-8	0.49	64	5.6
17	170	1.3	12	1.69	144	15.6
11	120	-4.7	-38	22.09	1444	178.6
19	170	3.3	12	10.89	144	39.6
18	180	2.3	22	5.29	484	50.6
17	160	1.3	2	1.69	4	2.6
15	140	-0.7	-18	0.49	324	12.6
17	180	1.3	22	1.69	484	28.6
18	190	2.3	32	5.29	1024	73.6

	x	y	(x-)2	(y-)2	(x-)(y-)
Total sum	157	1580	49.61	5560	407.4
Mean	15.7	158

Therefore,

Sample size , n =10

Here, x̅ = (Σx / n)=15.7 ,

ȳ = (Σy/n)=158

SSxx = Σ(x-x̅)² = 49.61   

SSxy= Σ(x-x̅)(y-ȳ) = 407.4   

Estimated slope ,

ß1 = SSxy/SSxx

= 407.4/49.61

=8.212

Intercept, ß0 = y̅-ß1 x̄

=158-(8.212)(15.7)

=158-128.928

=29.072

So, regression line is

Ŷ = 29.072 + 8.212 x

(b).

Calculate the sum of squared residuals, the total sum of squares, and the coefficient of determination:

SSE=Sum of squaures of residuals

= (SSxx * SSyy - SS²xy)/SSxx

=(49.61 5560-(407.4)2)/(49.61)

=(2756331.6-165974.76 )/49.61

=2590356.84/49.61

SSE= 52214.409

Total sum of squares=SST

=SSyy=5560

R² =(SST-SSE)/SST

=(5560-52214.409)/5560

=-8.39

(c).

Calculate the standard error of the estimate:

Standard error (SE) = √(SSE/(n-2))

=√(52214.409)/(9)

=√5801.601

=76.168

(d).

Calculate the standard error for the regression slope:

Estimated std error of slope =SE(ß1)

= SE/√Sxx

=76.168/√49.61

=76.168/7.043

=10.815

(e).

Conduct the hypothesis test to determine whether the regression slope coefficient is equal to 0:

Test using alpha α equals=0.10.

Ho: ß1= 0
H1: ß1╪ 0
n= 10
Alpha(α) = 0.10   

Estimated std error of slope =SE(ß1)

= SE/√Sxx

=76.168/√49.61

=76.168/7.043

=10.815

t -stat = Estimated slope/std error

=ß1 /SE(ß1)

=8.212 /10.815

=0.759

Degree of freedom ,df = n-2=8

p-value = 0.4696

Decison : p-value > α ,

Fail to reject Ho

Conclusion:

Fail to Reject Ho and conclude that slope is not significantly different from zero.