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 0000

Take a shot, what do you think?

This is an easy question, just calculate the means of for each of the three vehicles.

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? Crash1

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

Since the p-value > 0.05 we fail to reject H0 and conclude that the means are the same.

To determine if the mean crash damages are the same for the three vehicles (Goliath, Varmint, and Weasel), we can perform an analysis of variance (ANOVA) test. ANOVA is a statistical test used to compare means between three or more groups.

Here's how you can perform the ANOVA test to answer the research question:

1. Organize the data: Take the crash damage values and assign them to their respective groups - Goliath, Varmint, and Weasel. Since each group has five observations, you will have three sets of five crash damage values.

Goliath: 1600, 760, 880, 1950, 1220
Varmint: 1290, 1400, 1390, 1850, 950
Weasel: 1090, 2100, 1830, 1250, 0

2. Calculate the group means: Find the mean crash damage for each group. Add up the values in each group and divide by the number of observations in that group.

Goliath mean = (1600 + 760 + 880 + 1950 + 1220) / 5
Varmint mean = (1290 + 1400 + 1390 + 1850 + 950) / 5
Weasel mean = (1090 + 2100 + 1830 + 1250 + 0) / 5

3. Calculate the total mean: Find the mean crash damage across all groups. Add up all the crash damage values and divide by the total number of observations (15 in this case).

Total mean = (sum of all crash damage values) / (total number of observations)

4. Calculate the sum of squares: Calculate the sum of squares within groups (SS_within) and sum of squares between groups (SS_between).

SS_within measures the variation within each group:
- For each group, subtract the group mean from each individual value, square the result, and sum up these squared differences.

SS_between measures the variation between the group means:
- Subtract the total mean from each group mean, square the result, multiply it by the number of observations in that group, and sum up these squared differences.

5. Calculate degrees of freedom: Calculate the degrees of freedom for within groups (df_within) and between groups (df_between).

df_within = (number of groups) × (number of observations per group - 1)
df_between = (number of groups - 1)

6. Calculate mean squares: Divide the sum of squares within groups by the degrees of freedom within groups to get the mean square within groups (MS_within). Divide the sum of squares between groups by the degrees of freedom between groups to get the mean square between groups (MS_between).

MS_within = SS_within / df_within
MS_between = SS_between / df_between

7. Calculate the F-statistic: Divide the mean square between groups by the mean square within groups to get the F-statistic.

F-statistic = MS_between / MS_within

8. Determine the critical value: Compare the F-statistic to the critical value from the F-distribution table for your chosen significance level and degrees of freedom (alpha and df_between, df_within). If the F-statistic is greater than the critical value, we reject the null hypothesis of equal means.

9. Interpret the result: If the F-statistic exceeds the critical value, we can conclude that there is a significant difference in the mean crash damages between the three vehicles. If the F-statistic does not exceed the critical value, we fail to reject the null hypothesis, indicating that there is not enough evidence to suggest a difference in the means.

By following these steps and performing the ANOVA test, you can determine if the mean crash damages are the same for the Goliath, Varmint, and Weasel vehicles.