Question

\

J oouury 21 Dy 4:15 pm - Let r and Y equal the ACT scores in social science and natural science, respectively, for a student who is applying for admission to a small liberal arts college. A sample of n = 15 such students yielded the following data: Ya Yx Y 32 28 30 27 26 32 23 25 17 23 16 22 23 24 20 30 21 28 23 32 17 18 24 31 26 31 18 18 30 26 • Find point estimates for a, B and o2. • Calculate the fitted least squares regression line for these data. • Find residuals and make a residual plot. • Based on the residual plot, is the least squares regression line adequate for the dataset? Explain your answer.

Please do only the following: 1) Find residuals and make a residual plot. Please explain ALL steps and be NEAT.I am trying to learn this difficult material.

Answer 1

Excel sheet: Formula

			Sxx	Syy	Sxy			SSE	Residual
	X	Y	(X-Xbar)2	(Y-Ybar)2	(X-Xbar)(Y-Ybar)	Y_hat	(Y_hat-Ybar)2	(Y-Y_hat)2	Y-Y_hat
	32	28	=(B3-$B$19)^2	=(C3-$C$19)^2	=(B3-$B$19)*(C3-$C$19)	=$B$21+$B$20*B3	=(G3-$C$19)^2	=(C3-G3)^2	=C3-G3
	23	25	=(B4-$B$19)^2	=(C4-$C$19)^2	=(B4-$B$19)*(C4-$C$19)	=$B$21+$B$20*B4	=(G4-$C$19)^2	=(C4-G4)^2	=C4-G4
	23	24	=(B5-$B$19)^2	=(C5-$C$19)^2	=(B5-$B$19)*(C5-$C$19)	=$B$21+$B$20*B5	=(G5-$C$19)^2	=(C5-G5)^2	=C5-G5
	23	32	=(B6-$B$19)^2	=(C6-$C$19)^2	=(B6-$B$19)*(C6-$C$19)	=$B$21+$B$20*B6	=(G6-$C$19)^2	=(C6-G6)^2	=C6-G6
	26	31	=(B7-$B$19)^2	=(C7-$C$19)^2	=(B7-$B$19)*(C7-$C$19)	=$B$21+$B$20*B7	=(G7-$C$19)^2	=(C7-G7)^2	=C7-G7
	30	27	=(B8-$B$19)^2	=(C8-$C$19)^2	=(B8-$B$19)*(C8-$C$19)	=$B$21+$B$20*B8	=(G8-$C$19)^2	=(C8-G8)^2	=C8-G8
	17	23	=(B9-$B$19)^2	=(C9-$C$19)^2	=(B9-$B$19)*(C9-$C$19)	=$B$21+$B$20*B9	=(G9-$C$19)^2	=(C9-G9)^2	=C9-G9
	20	30	=(B10-$B$19)^2	=(C10-$C$19)^2	=(B10-$B$19)*(C10-$C$19)	=$B$21+$B$20*B10	=(G10-$C$19)^2	=(C10-G10)^2	=C10-G10
	17	18	=(B11-$B$19)^2	=(C11-$C$19)^2	=(B11-$B$19)*(C11-$C$19)	=$B$21+$B$20*B11	=(G11-$C$19)^2	=(C11-G11)^2	=C11-G11
	18	18	=(B12-$B$19)^2	=(C12-$C$19)^2	=(B12-$B$19)*(C12-$C$19)	=$B$21+$B$20*B12	=(G12-$C$19)^2	=(C12-G12)^2	=C12-G12
	26	32	=(B13-$B$19)^2	=(C13-$C$19)^2	=(B13-$B$19)*(C13-$C$19)	=$B$21+$B$20*B13	=(G13-$C$19)^2	=(C13-G13)^2	=C13-G13
	16	22	=(B14-$B$19)^2	=(C14-$C$19)^2	=(B14-$B$19)*(C14-$C$19)	=$B$21+$B$20*B14	=(G14-$C$19)^2	=(C14-G14)^2	=C14-G14
	21	28	=(B15-$B$19)^2	=(C15-$C$19)^2	=(B15-$B$19)*(C15-$C$19)	=$B$21+$B$20*B15	=(G15-$C$19)^2	=(C15-G15)^2	=C15-G15
	24	31	=(B16-$B$19)^2	=(C16-$C$19)^2	=(B16-$B$19)*(C16-$C$19)	=$B$21+$B$20*B16	=(G16-$C$19)^2	=(C16-G16)^2	=C16-G16
	30	26	=(B17-$B$19)^2	=(C17-$C$19)^2	=(B17-$B$19)*(C17-$C$19)	=$B$21+$B$20*B17	=(G17-$C$19)^2	=(C17-G17)^2	=C17-G17
Total	=SUM(B3:B17)	=SUM(C3:C17)	=SUM(D3:D17)	=SUM(E3:E17)	=SUM(F3:F17)		=SUM(H3:H17)	=SUM(I3:I17)
Mean	=B18/15	=C18/15		=(C19-$C$19)^2
b	=F18/D18
a	=C19-B20*B19
SSR	=H18
SST	=B22+B24
SSE	=I18		MSE	=B24/13	sigma2	=E24^2
R2	=1-(I18/E18)

Excel sheet : values

			Sxx	Syy	Sxy			SSE	Residual
	X	Y	(X-Xbar)2	(Y-Ybar)2	(X-Xbar)(Y-Ybar)	Y_hat	(Y_hat-Ybar)2	(Y-Y_hat)2	Y-Y_hat
	32	28	79.80444	2.777778	14.88889	30.85506	20.44603	8.151377	-2.85506
	23	25	0.004444	1.777778	0.088889	26.29959	0.001139	1.688932	-1.29959
	23	24	0.004444	5.444444	0.155556	26.29959	0.001139	5.28811	-2.29959
	23	32	0.004444	32.11111	-0.37778	26.29959	0.001139	32.49468	5.700411
	26	31	8.604444	21.77778	13.68889	27.81808	2.204472	10.12462	3.18192
	30	27	48.07111	0.444444	4.622222	29.84273	12.3159	8.081139	-2.84273
	17	23	36.80444	11.11111	20.22222	23.26261	9.429358	0.068963	-0.26261
	20	30	9.404444	13.44444	-11.2444	24.7811	2.409434	27.23694	5.218902
	17	18	36.80444	69.44444	50.55556	23.26261	9.429358	27.69504	-5.26261
	18	18	25.67111	69.44444	42.22222	23.76877	6.57698	33.27872	-5.76877
	26	32	8.604444	32.11111	16.62222	27.81808	2.204472	17.48846	4.18192
	16	22	49.93778	18.77778	30.62222	22.75644	12.79414	0.572207	-0.75644
	21	28	4.271111	2.777778	-3.44444	25.28726	1.094266	7.358948	2.712738
	24	31	0.871111	21.77778	4.355556	26.80575	0.22318	17.59171	4.194247
	30	26	48.07111	0.111111	-2.31111	29.84273	12.3159	14.76661	-3.84273
Total	346	395	356.9333	303.3333	180.6667		91.44689	211.8864
Mean	23.06667	26.33333		0
b	0.506164
a	14.65783
SSR	91.44689
SST	303.3333
SSE	211.8864		MSE	16.29896	sigma2	265.656
R2	0.301473

a)

a = 14.65783

B=0.506164

02 = 265.656

b)

û = 14.65783 +0.506161.6

c) Residuals are in table (SSE)

X Variable 1 Residual Plot Residuals 10 20 40 X Variable 1

X Variable 1 Line Fit Plot Predicted Y 10 30 40 20 X Variable 1

d) Since $R^2$ is only 30% and from the above residual plot and line fit plot, we may conclude that the model is not adequate.