Question

Question 4 Researchers studied four different blends of gasoline to determine their effect on miles per gallon (MPG) of a car. An experiment was conducted with a total of 28 cars of the same type, model, and engine size, with 7 cars randomly assigned to each treatment group. The gasoline blends are referred to as A,B,C, and D.The MPGs are shown below in the table Gasoline Miles Per Blend Gallon A 26 28 29 23 24 25 26 B 27 29 31 32 25 24 28 C 29 31 32 34 24 28 27 D 30 31 37 38 36 35 29 We want to test the null hypothesis that the four treatment groups have the same mean MPG vs. the alternative hypothesis that not all of the means are equal. a) Before carrying out the analysis, check the validity of any assumptions necessary for the analysis you will be doing. Write a brief statement of your findings b) Test the null hypothesis that the four gasoline blends have the same mean MPGs, i.e., Test Ho: ua=ub=uc=ud vs. the alternative hypothesis Ha: not all the means are equal. c) If your hypothesis test in (b) indicates a significant difference among the treatment groups, conduct pairwise multiple comparison tests on the treatment group means. Underline groups of homogeneous means. d) Briefly state your conclusions. ( Use IBM SPSS for all calculations)

Answer 1

a)

i) The data provided is measured at ratio level.

ii) Here we have four categorical independent groups A, B, C and D.

iii) The observations are independent, the observations are made on 28 different subjects which are randomly selected under each category.

iv) Equality of variances assumption

v)

Test for Equal Variances: A, B, C, D

Method

Null hypothesis	All variances are equal
Alternative hypothesis	At least one variance is different
Significance level	α = 0.05

Tests

Method	Test Statistic	P-Value
Levene	0.81	0.501

Test for Equal Variances: A, B, C, D

Since p-value is more than level of significance we fail to reject null hypothesis and we conclude that the variances are equal.

b)

One-way ANOVA: A, B, C, D

Method

Null hypothesis	All means are equal
Alternative hypothesis	Not all means are equal
Significance level	α = 0.05

Equal variances were assumed for the analysis.

Factor Information

Factor	Levels	Values
Factor	4	A, B, C, D

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Factor	3	231.0	77.000	8.19	0.001
Error	24	225.7	9.405
Total	27	456.7

Since p-value is less than level of significance we reject null hypothesis and we conclude that there is a significant differences between the groups of gasoline.

c) Tukey Pairwise Comparisons

Grouping Information Using the Tukey Method and 95% Confidence

Factor	N	Mean	Grouping
D	7	33.71	A
C	7	29.29	A	B
B	7	28.00		B
A	7	25.857		B

Means that do not share a letter are significantly different.

Tukey Simultaneous Tests for Differences of Means

Difference of Levels	Difference of Means	SE of Difference	95% CI	T-Value	Adjusted P-Value
B - A	2.14	1.64	(-2.38, 6.66)	1.31	0.567
C - A	3.43	1.64	(-1.09, 7.95)	2.09	0.184
D - A	7.86	1.64	(3.34, 12.38)	4.79	0.000
C - B	1.29	1.64	(-3.23, 5.81)	0.78	0.861
D - B	5.71	1.64	(1.19, 10.23)	3.49	0.010
D - C	4.43	1.64	(-0.09, 8.95)	2.70	0.056

.

Groups D-A and D-B are significantly different. Rest are not significantly different since zero is included in their interval.

d) d) From ANOVA we concluded that there exist a significant difference between the drugs, but from Tukey's simultaneous intervals we concluded that the gasoline D, A and B are significantly different.