Question

An engineer is studying the effect of cutting speed on the rate of metal removal in a machining operation. However, the rate of metal removal is also related to the hardness of the test specimen. Five observations are taken at each cutting speed. The amount of metal removed y and the hardness of the specimen x are shown below in the format of a data file. Column 1 has treatment (1 for cutting speed 1000, 2 for cutting speed 1200, 3 for cutting speed 1400), column 2 has x=hardness, column 3 has y=amount of metal removed.
1 125 77.4
1 120 68.4
1 140 90.4
1 150 97.9
1 136 87.6
2 133 85.4
2 140 94.4
2 125 74
2 120 64.8
2 165 112.1
3 130 79.6
3 175 117.6
3 132 82.3
3 141 91.9
3 124 72.9


Do an analysis of these data and include the following.
1. A scatterplot of the ys versus the xs, using different symbols (or colours) to distinguish the points corresponding to different cutting speeds. Do the ys appear to be related to the xs? Include regression lines of y versus x for each cutting speed. Does it seem reasonable to believe that these three lines have the same slope?
2. An ANCOVA table.
3. An ANOVA table based on the ys, disregarding the xs, to determine whether there are differences in the (mean) amount of metal removed for different cutting speeds. Comment on differences with part (2).

You will be asked a few questions concerning the analysis.

Please use 3 decimal places for the answers below which are not integer-valued.


Part a)
The type II SS for cutting speed (treatment) is______ and its degree of freedom is______    

Part b)
The MSE for ancova is:______    and its degree of freedom is______    

Part c)
The appropriate F ratio for cutting speed is:
______  ​

Part d)
What is the estimate slope for x=hardness?
______  

Answer 1

MINITAB used.

Do an analysis of these data and include the following.
1. A scatterplot of the ys versus the xs, using different symbols (or colours) to distinguish the points corresponding to different cutting speeds. Do the ys appear to be related to the xs? Include regression lines of y versus x for each cutting speed. Does it seem reasonable to believe that these three lines have the same slope?

Yes, it appear that y related to the x. It seems reasonable to believe that these three lines have the same slope.

2. An ANCOVA table.

Source	DF	Adj SS	Adj MS	F-Value	P-Value
x	1	2993.47	2993.47	315.99	0.000
Group	2	3.90	1.95	0.21	0.817
Error	11	104.21	9.47
Total	14	3149.46

3. An ANOVA table based on the ys, disregarding the xs, to determine whether there are differences in the (mean) amount of metal removed for different cutting speeds. Comment on differences with part (2).

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Group	2	51.78	25.89	0.10	0.905
Error	12	3097.68	258.14
Total	14	3149.46

Calculate F=0.10, P=0.905 which is > 0.05 level. Ho is not rejected.

You will be asked a few questions concerning the analysis.

Please use 3 decimal places for the answers below which are not integer-valued.


Part a)
The type II SS for cutting speed (treatment) is 3.903 and its degree of freedom is 2

Part b)
The MSE for ancova is:   9.473 and its degree of freedom is 11

Part c)
The appropriate F ratio for cutting speed is:
0.206

Part d)
What is the estimate slope for x=hardness?

0.9302

MINITAB ouput for ANCOVA

General Linear Model: y versus x, Group

Method

Factor coding

(-1, 0, +1)

Factor Information

Factor	Type	Levels	Values
Group	Fixed	3	1, 2, 3

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
x	1	2993.47	2993.47	315.99	0.000
Group	2	3.90	1.95	0.21	0.817
Error	11	104.21	9.47
Total	14	3149.46

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
3.07788	96.69%	95.79%	93.06%

Coefficients

Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	-41.06	7.22	-5.69	0.000
x	0.9302	0.0523	17.78	0.000	1.03
Group
1	0.56	1.13	0.49	0.631	1.36
2	0.13	1.12	0.11	0.912	1.33

Scatterplot of y vs x Group 120 110 100 90 80 70 60 170 180 130 140 150 160 120