Question

Engineers are testing company fleet vehicle fuel economy (miles per gallon) performance by using different types of fuel. One vehicle of each size is tested. Does this sample provide sufficient evidence to conclude that there is a significant difference in treatment means?

	87 Octane	89 Octane	91 Octane	Ethanol 5%	Ethanol 10%
Compact	30.8	28.4	17.7	30.7	31.1
Mid-Size	17.0	19.9	20.1	17.1	31.4
Full-Size	28.9	29.0	20.4	27.0	31.3
SUV	21.9	22.8	19.5	18.7	29.6

(b) Fill in the boxes. (Round your SS values to 3 decimal places, F values to 2 decimal places, and other answers to 4 decimal places.)

Two-Factor ANOVA
Source	SS	df	MS	F	p-value
Treatments (Fuel Type)
Blocks (Vehicle Size)
Error
Total

Group	Mean	n	Std. Dev
87 Octane
89 Octane
91 Octane
Ethanol 5%
Ethanol 10%
Compact
Mid-Size
Full-Size
SUV
Total

Answer 1

Null Hypothesis:

: There is no difference among the fuel type in terms of fuel economy

: There is a difference amongst at least one pair

: There is no difference among the Vehicle types in terms of Fuel economy

: There is a difference amongst at least one pair

Level of significance:

Let us now form this table for calculating Totals and the Sum of squares.

	87 Octane	89 Octane	91 Octane	Ethanol 5%	Ethanol 10%	Total
Compact	30.8	28.4	17.7	30.7	31.1	138.7
Mid-Size	17	19.9	20.1	17.1	31.4	105.5
Full-Size	28.9	29	20.4	27	31.3	136.6
SUV	21.9	22.8	19.5	18.7	29.6	112.5
Total	98.6	100.1	77.7	93.5	123.4	493.3

Grand total (GT) =sum of all observation=493.3, Number of observation(n)=20, Number of treatments(t)=5 and Number of blocks(b)=4

1. Correction factor :

2.Total Sum of squares(TSS):

To get the sum of squares of individual values, we form the table of Squares below and add all values:

	87 Octane	89 Octane	91 Octane	Ethanol 5%	Ethanol 10%	Total
Compact	948.64	806.56	313.29	942.49	967.21	3978.19
Mid-Size	289	396.01	404.01	292.41	985.96	2367.39
Full-Size	835.21	841	416.16	729	979.69	3801.06
SUV	479.61	519.84	380.25	349.69	876.16	2605.55
Total	2552.46	2563.41	1513.71	2313.59	3809.02	12752.19

3.Treatment Sum of squares(Trss):

Where is the treatment total for treatment.

4. Block sum of squares: (BSS):

Where is the treatment total for Block

5. Error SS(ESS):

Source	SS	df	MS	F	p-value
Treatments(Fuel Type)	169.906	3	56.5018	4.66	0.0221
Blocks(Vehicle Size)	270.023	4	67.5058	5.57	0.0090
Error	145.417	12	12.1181
Total	584.946	19

Since p-value for treatments<0.05, this sample provide sufficient evidence to conclude that there is a significant difference in treatment means.

Group	Mean	n	Stdev
87 Octane	24.65	4	6.3763
89 Octane	25.025	4	4.4124
91 Octane	19.425	4	1.2093
Ethanol 5%	23.375	4	6.5327
Ethanol 10%	30.85	4	0.8426
Compact	27.74	5	2.2361
Mid-Size	21.1	5	2.2361
Full-Size	27.32	5	2.2361
SUV	22.5	5	2.2361
Total	20.55	20	5.5486