The best way to test the equality of more than 2 group means is ANOVA(Analysis of Variance) test.
Step 1: Stating the null and alternative hypothesis
, The means are not equal
Step 2 : Level of significance
Step 3 : Test Statistic
F statistic for ANOVA
where , F = MSB/MSE
Step 4 : Computing the test statistic
First , we need to compute the group means
Line U | Line M | Line N | Line L | |
n | 5 | 5 | 5 | 5 |
Group sum | 122 | 62 | 174 | 61 |
Group mean | 24.4 | 12.4 | 34.8 | 12.2 |
The overall mean = (122+62+174+61)/20 = 419/20 = 20.95
Now we need to compute SSB and SSE
where , SSB = Sum of Squares between the groups , and
SSE = Sum of Squares within the groups
= 5*(24.4 - 20.95)2 + 5*(12.4 - 20.95)2 + 5*(34.8 - 20.95)2 + 5*(12.2 - 20.95)2
= 59.5125 + 365.5125 + 959.1125 + 382.8125
= 1766.95
Now , computing SSE for each group
SSE for group Line U ,where
Line U | ||
25 | 0.6 | 0.36 |
22 | -2.4 | 5.76 |
29 | 4.6 | 21.16 |
11 | -13.4 | 179.56 |
35 | 10.6 | 112.36 |
Total | 319.2 |
Similarly , for all other groups
SSE for group Line M , where
Line M | ||
15 | 2.6 | 6.76 |
12 | -0.4 | 0.16 |
11 | -1.4 | 1.96 |
22 | 9.6 | 92.16 |
2 | -10.4 | 108.16 |
Total | 209.2 |
SSE for group Line N ,
Line N | ||
35 | 0.2 | 0.04 |
38 | 3.2 | 10.24 |
41 | 6.2 | 38.44 |
15 | -19.8 | 392.04 |
45 | 10.2 | 104.04 |
Total | 544.8 |
SSE for group L , where
Line L | ||
9 | -3.2 | 10.24 |
5 | -7.2 | 51.84 |
2 | -10.2 | 104.04 |
35 | 22.8 | 519.84 |
10 | -2.2 | 4.84 |
Total | 690.8 |
SSE = Sum of all the SSE for each groups = 319.2 + 209.2 + 544.8 + 690.8 = 1764
constructing the ANOVA table
SS | DF | MS | F | |
SSB | 1766.95 | 4-1=3 | 1766.95/3 = 588.983 | 588.983/110.25 = 5.342 |
SSE | 1764 | 20-4=16 | 1764/16 = 110.25 | |
Total | 3530.95 | 20-1 = 19 1 |
Step 5 : Decision
Critical value of F at (16,3) for = 0.05 = 3.24
Calculated value of F = 5.342
Since Calculated value (5.342) > Critical Value (3.24)
We reject Null hypothesis(H0)
i.e we don't have significant evidence to show that 4 bus lines have same mean of number of passengers on it at different random times of day given the sample data from one day .
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Use =0.05 to test if these 4 Bus lines has a same average (mean) # of...
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