It is of interest to determine whether there is any significant difference in mi
ID: 3327117 • Letter: I
Question
It is of interest to determine whether there is any significant difference in mileage per gallon between gasolines A, B, C, D and E. Design an experiment that uses five different drivers, five different cars, and five different roads (alfa,beta....).
Suppose that in carrying out the experiment in (a), the numbers of miles per gallon are as given in the following table. Write all of the hypotheses to determine whether there are any differences at the -significance level.
DRIVERS
CARS
15
16
14
18
13
17
18
17
16
17
12
14
11
18
15
13
16
15
15
18
16
15
16
11
19
Show your calculations and fill in the table in the following:
Source of variation
Amount of Variation
df
Mean square
F
Prepare the data given in (b) to obtain the results from MINITAB. Which way do you follow to obtain the numerical results of Graeco Latin Squares Design in MINITAB?
DRIVERS
CARS
15
16
14
18
13
17
18
17
16
17
12
14
11
18
15
13
16
15
15
18
16
15
16
11
19
Explanation / Answer
Design of experiment that uses five different drivers, five different cars, and five different roads is Graeco Latin Square Design of order 5 ANSWER
ANOVA
Since the given data does not indicate the treatment gasoline and the road used, the given data is not analysed as Graeco Latin Squares Design. The analysis is that of RBD (Randomised Block Design) i.e., two-way ANOVA with one observation per cell.
Back-up Theory
Suppose we have data of a 2-way classification ANOVA, with r rows, c columns and 1 observation per cell.
Let xij represent the observation in the ith row-jth column, i = 1,2,……,r ; j = 1,2,…..,c.
Then the ANOVA model is: xij = µ + i + j + ij, where µ = common effect, i = effect of ith row, j = effect of jth column, and ijk is the error component which is assumed to be Normally Distributed with mean 0 and variance 2.
Now, to work out the solution,
Terminology:
Row total = xi.= sum over j of xij
Column total = x.j = sum over i of xij
Grand total = G = sum over i of xi. = sum over j of x.j
Correction Factor = C = G2/N, where N = total number of observations = r x c
Total Sum of Squares: SST = (sum over i,j of xij2) – C
Row Sum of Squares: SSR = {(sum over i of xi.2)/(c)} – C
Column Sum of Squares: SSC = {(sum over j of x.j2)/(r)} – C
Error Sum of Squares: SSE = SST – SSR - SSC
Mean Sum of Squares = Sum of squares/Degrees of Freedom
Degrees of Freedom:
Total: N (i.e., rc) – 1;
Rows: (r - 1);
Columns: (c - 1);
Error: DF for Total – DF for Rows – DF for Columns;
Fobs:
for Rows: MSSR/MSSE;
for Columns: MSSC/MSSE;
Fcrit: upper % point of F-Distribution with degrees of freedom n1 and n2, where n1 is the DF for the numerator MSS and n2 is the DF for the denominator MSS of Fobs
Significance: Fobs is significant if Fobs > Fcrit
Calculations: [Excel-based]
G
385
C
5929
SST
116
SSR
22.8
SSC
12.4
ANOVA TABLE
Source
DF
SS
MSS
Fobs
Fcrit
Row
4
22.8
5.7
1.128713
4.772578
Column
4
12.4
3.1
0.613861
4.772578
Error
16
80.8
5.05
Total
24
116
4.833333
Since both Fobs are less than the respective Fcrit, no effect is significant.
=> there is no evidence to suggest that neither driver effect nor car effect exists. DONE
G
385
C
5929
SST
116
SSR
22.8
SSC
12.4
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