Final answer:
The statistical technique to identify mutually exclusive groups is cluster analysis. Tests of independence like the chi-square test are used to determine if there is a relationship between two variables. For comparing means of different groups, an Analysis of Variance (ANOVA) test is used.
Step-by-step explanation:
The statistical technique used to identify mutually exclusive groups is called cluster analysis. This method organizes data into groups based on similarities and differences in the data. Mutual exclusivity means that elements can only belong to one group and not multiple groups. When terms such as 'mutually exclusive' and 'independent' are mentioned in statistical testing, they refer to specific properties of statistical events. Mutual exclusivity in probability theory means that two events cannot occur at the same time. For example, when looking at two separate events, D and E, if they share no possible outcomes in common, they are considered mutually exclusive. This is often illustrated with a statement such as “P(A AND C) = 0” indicating that the probability of events A and C occurring together is zero.
In statistics, we use tests of independence like the chi-square test of independence to determine if there is a relationship between two categorical variables. The null hypothesis typically states that there is no relationship (independence) between the variables. Chi-square compares the observed counts in categories to the counts we would expect if the variables were indeed independent of each other.
To address the provided scenarios: for the collaborative exercise, an Analysis of Variance (ANOVA) test would be used at the 1 percent level of significance to analyze if there are differences in the means of the number of states visited among the four age and gender groups. For polling of political opinions within married couples, one might question whether the opinions are independent or matched (paired), and answering this would require understanding the relationship between the spouses' responses. In summary, cluster analysis is used for identifying mutually exclusive groups, while chi-square and ANOVA tests assess independence and differences between group means, respectively.