data consists of one independent variable X, which has 5 levels (i.e. X = 1, 2, 4, 5, 6) and a numeric dependent variable Y

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1.    The data consists of one independent variable X, which has 5 levels (i.e. X = 1, 2, 4, 5, 6) and a numeric dependent variable Y . Variable X2 and X3 are the square (X2 = X2) and cube (X3 = X3) of X, respectively. Below are part of the SAS code and outputs of fitting a SLR, quadratic regression

and cubic regression models of Y on X.

 PROC  REG DATA=data;

MODEL Y=X; title ’Simple linear regression model’; RUN; PROC REG DATA=data;

MODEL Y=X X2; title ’Quadratic regression model’; RUN; PROC REG DATA=data;

MODEL Y=X X2 X3; title ’Cubic regression model’; RUN; PROC glm; class X;

model Y=X; title ’one-way ANOVA’; RUN;

Variable	DF	Estimate                  Error		t Value	Pr > |t|
Intercept	1	1.04319               1.97378		0.53	0.6060
X	1	-5.32626                1.34422		-3.96	0.0016
X2	1	0.96029               0.18745		5.12	0.0002
		Cubic regression model
		Analysis of Variance
		Sum of		Mean
Source		DF              Squares		Square	F  Value        Pr > F
Model		3           215.40920             71.80307			21.22	<.0001
Error		12             40.60830               3.38402
Corrected Total		15           256.01750
		Parameter Estimates
		Parameter           Standard
Variable	DF	Estimate                  Error       t Value			Pr >	|t|
Intercept	1	-4.35868	4.36821	-1.00	0.3381
X	1	1.45448	5.10018	0.29	0.7804
X2	1	-1.21839	1.59489	-0.76	0.4597
X3	1	0.20125	0.14637	1.37	0.1943

one-way ANOVA The GLM Procedure

Sum of

Source                      DF               Squares          Mean Square       F  Value        Pr > F 

Model	4	216.9241667	54.2310417	15.26	0.0002
Error	11	39.0933333	3.5539394
Corrected Total	15	256.0175000

a)    Use the SAS outputs above to find out a reasonable regression model of Y on X. Provide theappropriate test statistics, df and p-value to justify your choice and explain briefly.

 b)   Suppose we are ONLY interested three pairwise comparisons among X=1,5 and 6. That is 5-1 (which means µx=5 −µx=1), 6-1 and 6-5. Below are the Fisher’s LSD output for these three pairwise comparisons. Use the Bonferroni approach to test all three pairwise comparisons (i.e. 5-1, 6-1 and 6-5). Provide the Bonferroni confidence intervals numerically. Set the familywise error rate at 0.05. t-Tests (LSD) for Y

NOTE: This test controls the Type I comparisons error rate, not the familywise error rate.

Alpha                                                    0.05

Error  Degrees of Freedom                     11

Error  Mean Square (MSE)          3.553939

Critical  Value of t                    2.201

Comparisons significant at the 0.05 level are indicated by ***.

Difference 

X Comparison	Between Means	95% Confidence Limits
6 - 5	7.125	3.532	10.718 ***
6 - 1	7.908	4.739	11.077 ***
5 - 1	0.783	-3.004	4.571

 c)   We  can  also  apply  Tukey’s  HSD  approach  as  well  as  Scheff´e  approach  to  test  all  three  pairwise comparisons in part b (i.e. 5-1, 6-1 and 6-5). Compare the three approaches (i.e. Bonferroni, Tukey  and  Scheff´e),  and  find  out  which  one  is  the  best?   Explain  briefly.   Set  the  familywise  error rate at 0.05.

1.    An automobile manufacturer want to compare the gasoline consumption rate (miles per gallon) on five particular brands of cars. Five cars were used, one for each brand. Four drivers were randomly selected in this study. Each driver drove each car twice over a 25-mile test course and the miles  per gallon (mpg) were recorded. Therefore, 5 × 4 × 2 = 40 mpg values were recorded. A crude SAS analysis output is the following. Set α = 0.05.

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