Question

 

Consider the following data:

x 1 4 5 7 8 12 11 14 19 20

y 1 54 125 324 512 5,530 5,331 5,740 7,058 7,945

Use Excel to resolve:

a. Construct a scatter plot of the data. Determine the order of the polynomial that is represented by this data.

b. Obtain an estimate of the model identified in part a. c. Conduct a test of hypothesis to determine if a third- order, as opposed to a first-order, polynomial is a better representation of the relationship between y and x. Use a significance level of 0.05 and the p-value approach.

Answer 1

(a)

Following is the scatter plot of the data :

Scatter plot shows that relationship between the variables is not linear so third -order polynomial will be appropriate.

(b)

Following is the scatter model with fitted curve:

Following is the output of regression analysis:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.973907553
R Square	0.948495921
Adjusted R Square	0.922743882
Standard Error	921.2297353
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	3	93773686.65	31257895.55	36.83187605	0.000293031
Residual	6	5091985.352	848664.2253
Total	9	98865672
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	830.0930499	1273.111874	0.652018936	0.538548943	-2285.099476	3945.285575
X	-887.5683225	519.763671	-1.707638244	0.138570481	-2159.384206	384.2475613
X^2	164.8555061	58.41583815	2.822102897	0.030270332	21.91709968	307.7939125
X^3	-5.207654756	1.829764124	-2.846079824	0.029332028	-9.684926268	-0.730383245

(c)

The F test statistics is:

F = 36.83

The p-value is:

p-value = 0.0003

Since p-value is less than 0.05 so we can conclude that model is significant.