Final answer:
To compare the means of automotive groups and test for differences, statistical methods, and a computerized database or software package must be used. ANOVA tests for means and F-tests or Chi-square tests for variances and independence, respectively, would solve the problem. Box plots are useful for visual analysis of data.
Step-by-step explanation:
If we are to solve the mathematical problem completely, we must use statistical methods with the guidance of a computerized database or statistical software package. To calculate the means of the three groups in the values variables and compare them, one could use software like SPSS, SAS, Stata, R, or Python with relevant statistical packages. For each group (American, Japanese, and European car owners), the mean can be calculated by summing all the values and dividing by the number of observations in each group.
After calculating the means, we can conduct an Analysis of Variance (ANOVA) to see if there is a significant difference between the groups' values variables. If the F-statistic from the ANOVA is significant at a p-value less than 0.05 (for a 5 percent significance level), we can say that at least one group mean is statistically different from the others. It may then be useful to conduct a post-hoc test such as Tukey's HSD to determine which specific groups differ.
Regarding variance in mileage or car and family size independence, the same analytical approach can be used. For mileage variance between groups, a two-sample F-test for variances is appropriate, whereas for car and family size independence, a Chi-square test is recommended. Lastly, comparing distributions of car types between families and singles would also involve a Chi-square test for a given level of significance.
When determining the most likely group to have an outlier in the car purchasing age, box plots are insightful. The group with the longest whiskers or points beyond the whiskers is likely to have outliers. Comparing box plots can show differences between the central tendencies and variabilities of the different groups.