Question

Use the same data set for particle board stiffness as in question 2

e) Are the assumptions satisfied? Provide appropriate plots to prove it and comment what the plots show you.

f) Calculate a 95% confidence on the slop of the regression line and interpret what it means.

g) Obtain from Excel a 95% confidence interval on the average stiffness at a density of 8.5 and explain what the interval means.

h) Obtain from Excel a 95% prediction interval on the stiffness of a run of particle board for which its density is 7.8, and explain what the interval means.

Density	Stiffness	Temp
9.5	14.81	70.6
8.4	17.50	73.3
9.8	14.01	66.2
11	19.44	70.1
8.3	7.57	69.3
9.9	14.19	69.1
8.6	9.71	69.8
6.4	8.08	64.4
7	5.30	65.4
8.2	10.73	67.8
17.4	43.24	69.7
15	25.32	66.9
15.2	28.03	71.5
16.4	41.79	66.6
16.7	49.50	68.0
15.4	25.31	64.3
15	26.22	64.5
14.5	22.15	71.3
14.8	26.75	69.6
13.6	18.04	71.1
25.6	96.31	72.1
24.4	72.59	67.3
23.3	49.51	66.7
19.5	32.21	68.1
21.2	48.22	66.5
22.8	70.45	65.7
21.7	47.66	66.3
19.8	38.14	69.4
21.3	53.05	68.3

Answer 1

e) Are the assumptions satisfied? Provide appropriate plots to prove it and comment what the plots show you.

The regression has five key assumptions:

• Linear relationship
• Multivariate normality
• No or little multicollinearity
• No auto-correlation
• Homoscedasticity

From the plot, we can say that all the assumptions are met.

f) Calculate a 95% confidence on the slope of the regression line and interpret what it means.

The regression output is:

r²	0.845
	r	0.919
	Std. Error	8.869
	n	29
	k	1
	Dep. Var.	Stiffness
ANOVA table
Source	SS	df	MS	F	p-value
Regression	11,553.3915	1	11,553.3915	146.87	1.98E-12
Residual	2,123.9117	27	78.6634
Total	13,677.3032	28
Regression output					confidence interval
variables	coefficients	std. error	t (df=27)	p-value	95% lower	95% upper
Intercept	-21.5355
Density	3.5406	0.2922	12.119	1.98E-12	2.9412	4.1401

The 95% confidence on the slope of the regression line is between 2.9412 and 4.1401. It means that we are 95% confident that Stiffness is between 2.9412 and 4.1401 for each unit increase in Density.

g) Obtain from Excel a 95% confidence interval on the average stiffness at a density of 8.5 and explain what the interval means.

The Excel output is:

Predicted values for: Stiffness
		95% Confidence Interval		95% Prediction Interval
Density	Predicted	lower	upper	lower	upper	Leverage
8.5	8.55995	3.31266	13.80724	-10.37963	27.49953	0.083

The 95% confidence interval on the average stiffness at a density of 8.5 is between 3.31266 and 13.80724 for each unit increase in Density.

h) Obtain from Excel a 95% prediction interval on the stiffness of a run of particle board for which its density is 7.8, and explain what the interval means.

The Excel output is:

Predicted values for: Stiffness
		95% Confidence Interval		95% Prediction Interval
Density	Predicted	lower	upper	lower	upper	Leverage
7.8	6.08151	0.50665	11.65636	-12.95143	25.11444	0.094

The 95% prediction interval on the stiffness of a run of particle board for which its density is 7.8 is between -12.95143 and 25.11444 for each unit increase in Density.