Question

Part 1. Consider the dataset below. You will perform a series of regressions and data transformations. Be sure to keep a record of all your computer results. First, please perform a simple linear regression. Predict Y if X = 40. To avoid rounding errors in ALL your calculations, please perform your calculations on your spreadsheet referencing data from your regression output.

X	Y
54	6
42	16
28	33
38	18
25	41
70	3
48	10
41	14
20	45
52	9
65	5

		14.0363
		14.1891
		17.2164
		21.5627
		None of the above

Part 2

Please perform a polynomial regression of Y against X and X-squared.  What is the coefficient of the curvature term?

		92.8725
		-2.7222
		0.0208
		0.9851
		None of the above

Part 3.

With other OLS regression conditions satisfied, can we utilize the estimated equation of this model to predict Y?

		Yes. The regression is statistically significant and the coefficient of determination is reasonably high.
		No. Although the regression is significant, there is a high likelihood that multicollinearity is a problem due to the inclusion of the X-squared term.
		No. The regression model is nonlinear and cannot therefore be utilized to make a forecast.
		No. The standard error of the regression is too high, indicating that the unexplained variation (error term) exhibits heteroscedasticity.
		None of the above

Part 4.

Based on the parameter estimates of the quadratic model, predict Y if X = 40.

		14.0363
		14.1891
		17.2164
		21.5627
		19.1523

Part 5.

Perform a logarithmic transformation of only Y on the original dataset. That is, ln(Y) = B0 + B1(X) + e. Then predict Y if X = 40.

		21.5627
		17.2164
		2.7885

Answer 1

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.932145801
R Square	0.868895795
Adjusted R Square	0.854328661
Standard Error	5.64255772
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	1899.090245	1899.090245	59.64768355	2.92936E-05
Residual	9	286.5461186	31.83845762
Total	10	2185.636364
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	56.15733314	5.203079582	10.79309518	1.88916E-06	44.3871494
X	-0.8648668	0.111983087	-7.72319128	2.92936E-05	-1.118190142

part 1)

y^ = 56.1573 - 0.8649 *x

= 56.1573 - 0.8649 *40

= 21.5627

part 2)

Y	X	x^2
6	54	2916
16	42	1764
33	28	784
18	38	1444
41	25	625
3	70	4900
10	48	2304
14	41	1681
45	20	400
9	52	2704
5	65	4225

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.994003876
R Square	0.988043706
Adjusted R Square	0.985054633
Standard Error	1.807349945
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	2	2159.504253	1079.752	330.5518	2.04E-08
Residual	8	26.1321106	3.266514
Total	10	2185.636364
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	92.87251293	4.436918394	20.93176	2.85E-08	82.64096
X	-2.722212931	0.211088773	-12.8961	1.24E-06	-3.20898
x^2	0.020770253	0.002326226	8.928735	1.96E-05	0.015406

coefficient of the curvature term = 0.0208

part 3)

A) Yes. The regression is statistically significant and the coefficient of determination is reasonably high.

part 4)

17.21640086

part 5)

ln (Y)	Y	X
1.791759	6	54
2.772589	16	42
3.496508	33	28
2.890372	18	38
3.713572	41	25
1.098612	3	70
2.302585	10	48
2.639057	14	41
3.806662	45	20
2.197225	9	52
1.609438	5	65

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.991254363
R Square	0.982585213
Adjusted R Square	0.980650236
Standard Error	0.122438958
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	7.612614045	7.612614	507.8022	3.16E-09
Residual	9	0.134921686	0.014991
Total	10	7.747535731
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	4.978748604	0.112902636	44.09772	7.92E-12	4.723345	5.234152	4.723345	5.234152
X	-0.054757465	0.002429943	-22.5345	3.16E-09	-0.06025	-0.04926	-0.06025	-0.04926

ln y^ = 4.9787 -0.0547 * x

y^ =

16.25580413