Question

Hypothesis Testing

Please type the answer by computer. (must answer and step degrees of freedom)

State SSW, SSB, the degrees of freedom, the F-statistics, and the test result.
Three types of medium sized cars assembled in New Zealand have been test driven by a motoring magazine and compared on a variety of criteria. In the area of fuel efficiency performance, five cars of each brand were each test driven 1000 km; the km per litre data are obtained as follows: Mean 8.0 8.6 9.9 Total 40.0 43 49.5 Kilometres per litre 8.4 8.0 10.4 8.4 9.6 10.6 7.6 8.5 9.7 8.0 Brand A 7.6 Brand B 7.8 Brand C 9.6 9.2 The task is to determine if there is a difference in the mean fuel consumption for the three makes of car. State SSW, SSB, the degrees of freedom, the F-statistics, and the test result.

Answer 1

H0: Null Hypothesis: $\mu 1=\mu 2=\mu 3$

HA: Alternative Hypothesis: $\mu 1\neq \mu 2\neq \mu 3$ (At least one mean is different from the other 2 means)

From the given Table, the following Table is calculated:

	Brand A	Brand B	Brand C	Total
N	5	5	5	15
$\sum X$	40	43	49.5	132.5
Mean	40/5 =8	43/5 = 8.6	49.5/5=9.9	132.5/15=8.833
$\sum X^{2}$	320.64	372.06	491.41	1184.11
Std. Dev.	0.4	0.7517	0.5831	0.989

From the above Table, ANOVA Table is calculated as follows:

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Sum of Squares	F
Between Treatments	SSB=9.4333	$\nu 1$= 2	9.4333/2=4.7167	F - statistic = 4.7167/0.355=13.2864
Within treatments	SSW=4.26	$\nu 2$ = 12	4.26/12=0.355
Total	13.6933		14

F - statistic = 4.7167/0.355=13.2864

Degrees of freedom for numerator = $\nu 1$ = 2

Degrees of freedom for denominator = $\nu 2$ = 12

By Technology, P - value = 0.00009

Since P - value = 0.0009 is less than $\alpha$ = 0.05, the difference is significant. Reject null hypothesis.

Conclusion:

The data support the claim that there is a difference in the mean fuel consumption for the three makes of car.

Answers to questions asked:
(a) SSW=4.26

(b) SSB=9.4333

(c)

Degrees of freedom for numerator = $\nu 1$ = 2

Degrees of freedom for denominator = $\nu 2$ = 12

(d)

F - statistic =13.2864

(e) Test result:

The difference is significant. Reject null hypothesis.

Conclusion:

The data support the claim that there is a difference in the mean fuel consumption for the three makes of car.