Question

An article entitled “A Method for Improving the Accuracy of Polynomial Regression Analysis” in the Journal of Quality Technology (1971, pp. 149-155) reported the following data:

x	770	800	840	810	735	640	590	560
y	280	284	292	295	298	305	308	315

(a) Fit a second-order polynomial to these data. What is the fitted polynomial regression model?

For parts (b) and (c) below, specify the hypotheses, test statistics, and conclusions
(b) Test for significance of regression using α = 0.05.
(c) Test the hypothesis that β11 = 0 using α = 0.05, where β11 is the coefficient for x2 in the polynomial regression model.
(d) Compute the residuals from part (a) and use them to evaluate model adequacy.

Answer 1

a) The fitted polynomial regression model is

y = 561.9 - 0.6713 x + 0.000413 x^2

b)

Analysis of Variance

Source	DF	SS	MS	F	P
Regression	2	775.26	387.63	8.75	0.023
Residual Error	5	221.62	44.32
Total	7	996.87

The p-value for the corresponding F-test statistic is 0.023 and less than alpha = 0.05. Hence, we can conclude that at least one coefficient has a significant effect on the model.

c)

Predictor	Coef	SE Coef	T	P
Constant	561.9	174.3	3.22	0.023
x	-0.6713	0.5087	-1.32	0.244
x^2	0.000413	0.00036	1.14	0.308

The estimated p-value for the coefficient for x2 in the polynomial regression model is 0308 and greater than alpha=0.05. Hence, the coefficient is equal to zero and conclude that x2 does not significantly affect on Y at 0.05 level of significance.

d)

From the above plot and the estimated p-value for AD test, we can conclude that the residual is followed a normal distribution. Hence, the assumption of normality on error of the model is satisfied.