Question

In: Statistics and Probability

Engineers are testing company fleet vehicle fuel economy (miles per gallon) performance by using different types...

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

Solutions

Expert Solution

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

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