## Question 528:

1## Answer:

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Since we are comparing more than 2 means we can use the 1-Way Analysis of Variance. It is easiest to do this calculation using software and I've attached an excel file which shows all the calculations. I'll walk through the major steps here.

- First calculate the Grand Mean for all 15 values. We get 1432.
- Next we subtract the grand mean from each value and square the result. Then we add up all 15 squared differences to get the Total Sum of Squares = 2,520,640
- Next we subtract the mean for each autotype from the grand mean and square this difference. Then multiply this squared difference times the sample size for each group (all of them are 5). This is called the Sum of Squares Treatment and we add up all three to get 340,360.
- Now we calculate the Sum of Squares Error, which is the Total Sum of Squares minus the treatment sum of squares 2,520,640-340,360 = 2,180,280.
- The degrees of freedom for this test are 1 less than the number of groups (3-1) and 1 less than the total sample size (15-1).
- We divide the Sum of Squares Treatment by the Group degrees of freedom 340,360/2 = 170,180 which is called the Mean Square Treatment.
- We divide the Sum of Squares Error by the total degrees of freedom 2,180,280/14 = 181,690 called the Mean Square Error.
- Out test statistic F is the ration of the Mean Square Treatment by the Mean Square Error = 170,180/181690 = .93665.
- To find the p-value we can use the excel function =FDIST(.93665, 2,12) where the parameters are the f-statistics, the treatment degrees of freedom and the error degrees of freedom (14-2) and gets us a p-value of .4188.

Since the p-value of .4188 is well above .05 we would NOT reject the null hypothesis and conclude that there is no difference between the means crash damages.