In a bumper test, three types of autos were deliberately crashed into a barrier at 5 mph, and the resulting damage (in dollars) was estimated. Five test vehicles of each type were crashed, with the results shown below. Research question: Are the mean crash damages the same for these three vehicles?
Crash Damage ($)
Goliath Varmint Weasel
1,600 1,290 1,090
760 1,400 2,100
880 1,390 1,830
1,950 1,850 1,250
1,220 950 1,920
To determine whether the mean crash damages are the same for these three vehicles, we can use a statistical test called analysis of variance (ANOVA). ANOVA compares the mean differences between groups and determines if there is a statistically significant difference.
To conduct an ANOVA, we follow these steps:
Step 1: State the hypotheses
- Null hypothesis (H0): The mean crash damages are the same for the three vehicles.
- Alternative hypothesis (Ha): The mean crash damages are different for at least one of the vehicles.
Step 2: Calculate the grand mean
- Find the sum of all the crash damages for each vehicle and divide it by the total number of observations.
Step 3: Calculate the sum of squares (SS)
- Calculate the sum of squares for each vehicle and the total sum of squares.
Step 4: Calculate the mean sum of squares (MS)
- Calculate the mean sum of squares for between groups (MSB) and within groups (MSW).
Step 5: Calculate the F statistic
- Calculate F = MSB / MSW
Step 6: Find the critical value and make a decision
- With ANOVA, we need to compare the calculated F statistic to the critical value obtained from a table at a given significance level (e.g., α = 0.05).
If the calculated F statistic is greater than the critical value, we reject the null hypothesis and conclude that there are significant differences between the mean crash damages of the three vehicles.
For the provided data, perform the calculations mentioned above and compare the calculated F statistic to the critical value to determine whether the mean crash damages are the same for the three vehicles.