Question

A research firm tests the miles-per-gallon characteristics of three brands of gasoline. Because of different gasoline performance characteristics in different brands of automobiles, five brands of automobiles are selected and treated as blocks in the experiment; that is, each brand of automobile is tested with each type of gasoline. The results of the experiment (in miles per gallon) follow.

		Gasoline Brands
		I	II	III
Automobiles	A	18	21	20
	B	24	26	27
	C	30	28	34
	D	22	25	24
	E	20	23	24

(a)At α = 0.05, is there a significant difference in the mean miles-per-gallon characteristics of the three brands of gasoline?

State the null and alternative hypotheses.

H₀: At least two of the population means are equal.
H_a: At least two of the population means are different.

H₀: μ_I ≠ μ_II ≠ μ_III
H_a: μ_I = μ_II = μ_III    

H₀: μ_I = μ_II = μ_III
H_a: μ_I ≠ μ_II ≠ μ_III

H₀: Not all the population means are equal.
H_a: μ_I = μ_II = μ_III

H₀: μ_I = μ_II = μ_III
H_a: Not all the population means are equal.

Find the value of the test statistic. (Round your answer to two decimal places.)

=

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Do not reject H₀. There is sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.

Do not reject H₀. There is not sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.   

Reject H₀. There is not sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.

Reject H₀. There is sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.

(b) Analyze the experimental data using the ANOVA procedure for completely randomized designs. (Use α = 0.05.)

Find the value of the test statistic. (Round your answer to two decimal places.)

=

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Do not reject H₀. There is sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.

Reject H₀. There is not sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.   

Do not reject H₀. There is not sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.

Reject H₀. There is sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.

Compare your findings with those obtained in part (a).

The conclusion is the same as the conclusion in part (a).

The conclusion is different from the conclusion in part (a).    

What is the advantage of attempting to remove the block effect?

There is no advantage to removing the block effect because the conclusion is the same in either case.

We must remove the block effect in order to detect that there is a significant difference due to the brand of gasoline.    

We must remove the block effect in order to detect that there is no significant difference due to the brand of gasoline.

Answer 1

(a)At α = 0.05, is there a significant difference in the mean miles-per-gallon characteristics of the three brands of gasoline?

State the null and alternative hypotheses.

H₀: μ_I = μ_II = μ_III
H_a: Not all the population means are equal.

ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Automobiles	204.2667	4	51.06667	22.04317	0.000223	3.837853
Gasoline	22.8	2	11.4	4.920863	0.040421	4.45897
Error	18.53333	8	2.316667
Total	245.6	14

Find the value of the test statistic. (Round your answer to two decimal places.) F=4.9209

Find the p-value. (Round your answer to three decimal places.) p-value =0.0404

State your conclusion.

Reject H₀. There is sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.

(b) Analyze the experimental data using the ANOVA procedure for completely randomized designs. (Use α = 0.05.)

ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Gasoline Brands	22.8	2	11.4	0.614004	0.557341	3.885294
Error	222.8	12	18.56667
Total	245.6	14

Find the value of the test statistic. (Round your answer to two decimal places.) F=0.6140

Find the p-value. (Round your answer to three decimal places.) p-value=0.5573

State your conclusion.

Do not reject H₀. There is not sufficient evidence to conclude that the mean miles-per-gallon ratings for the three brands of gasoline are not all equal.

Compare your findings with those obtained in part (a).

The conclusion is different from the conclusion in part (a).    

What is the advantage of attempting to remove the block effect?

We must remove the block effect in order to detect that there is a significant difference due to the brand of gasoline.