Question

A statistical program is recommended.

Consider the following data for two variables, x and y.

x	22	24	26	30	35	40
y	12	20	33	35	40	36

(a) Develop an estimated regression equation for the data of the form

ŷ = b₀ + b₁x.  (Round b₀ to one decimal place and b₁ to three decimal places.

ŷ =

(b) Use the results from part (a) to test for a significant relationship between x and y. Use α = 0.05.

Find the value of the test statistic. (Round your answer to two decimal places.)

F =

Find the p-value. (Round your answer to three decimal places.)

p-value =

Is the relationship between x and y significant?

Yes, the relationship is significant.

No, the relationship is not significant.    

(c) Develop a scatter diagram for the data.

Does the scatter diagram suggest an estimated regression equation of the form

ŷ = b₀ + b₁x + b₂x²?

Explain.

Yes, the scatter diagram suggests that a linear relationship may be appropriate.

Yes, the scatter diagram suggests that a curvilinear relationship may be appropriate.  

  No, the scatter diagram suggests that a linear relationship may be appropriate.

No, the scatter diagram suggests that a curvilinear relationship may be appropriate.

(d) Develop an estimated regression equation for the data of the form

ŷ = b₀ + b₁x + b₂x².

(Round b₀ to one decimal place and b₁ to two decimal places and b₂ to four decimal places.)

ŷ =

(e) Use the results from part (d) to test for a significant relationship between x, x², and y. Use α = 0.05. Is the relationship between x, x², and y significant?

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

Is the relationship between x, x², and y significant?

Yes, the relationship is significant.

No, the relationship is not significant.    

(f) Use the model from part (d) to predict the value of y when x = 25.  (Round your answer to three decimal places.)

Answer 1

Output using excel:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.797173
R Square	0.635484
Adjusted R Square	0.544355
Standard Error	7.340818
Observations	6
ANOVA
	df	SS	MS	F	Significance F
Regression	1	375.7829	375.7829	6.973456	0.057536
Residual	4	215.5505	53.88761
Total	5	591.3333
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-7.61865	14.31041	-0.53239	0.622658	-47.3507	32.11343
x	1.25261	0.474342	2.64073	0.057536	-0.06438	2.569595

(a) Estimated regression equation :

ŷ = -7.6 + 1.253 x.

(b) Test statistic:

F = 6.97

p-value = 0.058

No, the relationship is not significant.    

(c) Scatter diagram for the data:

Yes, the scatter diagram suggests that a curvilinear relationship may be appropriate.  

------------------------

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.971161
R Square	0.943154
Adjusted R Square	0.905257
Standard Error	3.347372
Observations	6
ANOVA
	df	SS	MS	F	Significance F
Regression	2	557.7186	278.8593	24.88727	0.013553
Residual	3	33.61469	11.2049
Total	5	591.3333
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-170.727	41.00072	-4.16399	0.0252	-301.209	-40.2441
x	12.27742	2.744535	4.473405	0.020819	3.543081	21.01175
x2	-0.17813	0.044205	-4.02954	0.027474	-0.31881	-0.03745

(d) Develop an estimated regression equation for the data of the form

ŷ = -170.7 + 12.28 x + (-0.1781) x².

(e)

Test statistic:

F = 24.89

p-value = 0.013

Yes, the relationship is significant.

(f) Use the model from part (d) to predict the value of y when x = 25.  

ŷ = -170.7 + 12.28*25 + (-0.1781)*25² = 24.880