Question

In the production of printed circuit​ boards, errors in the alignment of electrical connections are a source of scrap. The accompanying table shows the registration error and the temperature used in the production of circuit boards in an experiment in which higher cost material was used.

Registration Error		Temperature
−10.7		278
3.7		320
3.2		299
10.6		321
−13.8		271
6.2		293
13.7		324

A. Fit a quadratic regression model and state the quadratic regression.

B. Perform a residual analysis on the results and determine if the regression model is valid.

C. At the 0.05 level of​ significance, is there a significant quadratic relationship between temperature and registration​ error?

D. At the 0.05 level of​ significance, determine whether the quadratic model is a better fit than the linear model.

E. Interpret the meaning of the coefficient of multiple determination.

F. Compute the adjusted r2.

Answer 1

(a) The quadratic regression model is:

y = -960.6411 + 6.0298*x - 0.0094*x²

(b) The residual plot is:

Since a random pattern of the residuals can be seen, the regression model is valid.

(c) The hypothesis being tested is:

H₀: β₁ = β₂ = 0

H₁: At least one β_i ≠ 0

The p-value is 0.0155.

Since the p-value (0.0155) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that there is a significant quadratic relationship between temperature and registration​ error.

(d) R² for the quadratic model is 0.876 while the R² for the linear model is 0.807. Since R² for the quadratic model is morel, it is a better fit than the linear model.

(e) 87.6% of the variation in the model is explained.

(f) Adjusted R² = 0.813

	R²	0.876
	Adjusted R²	0.813
	R	0.936
	Std. Error	4.474
	n	7
	k	2
	Dep. Var.	y
ANOVA table
Source	SS	df	MS	F	p-value
Regression	563.5158	2	281.7579	14.08	.0155
Residual	80.0614	4	20.0153
Total	643.5771	6
Regression output				confidence interval
variables	coefficients	std. error	t (df=4)	p-value	95% lower	95% upper
Intercept	-960.6411
x	6.0298	3.7799	1.595	.1859	-4.4650	16.5246
x²	-0.0094	0.0063	-1.481	.2126	-0.0269	0.0082