The following data represent political party by age from a...

The following data represent political party by age from a random sample of registered Iowa Voters $$ \begin{array}{lccccc} & \mathbf{1 7 - 2 9} & \mathbf{3 0 - 4 4} & \mathbf{4 5 - 6 4} & \mathbf{6 5 +} & \text { Total } \\ \hline \text { Republican } & 224 & 340 & 1075 & 561 & \mathbf{2 2 0 0} \\ \hline \text { Democrat } & 184 & 384 & 773 & 459 & \mathbf{1 8 0 0} \\ \hline \text { Total } & \mathbf{4 0 8} & \mathbf{7 2 4} & \mathbf{1 8 4 8} & \mathbf{1 0 2 0} & \mathbf{4 0 0 0} \\ \hline \end{array} $$ (a) Are the events "Republican" and "30-44" independent? Justify your answer. (b) Are the events "Democrat" and "65+" independent? Justify your answer. (c) Are the events "17-29" and "45-64" mutually exclusive? Justify your answer. (d) Are the events "Republican" and "45-64" mutually exclusive? Justify your answer.

The following data represent political party by age from a random sample of registered Iowa Voters
$$
\begin{array}{lccccc}
& \mathbf{1 7 - 2 9} & \mathbf{3 0 - 4 4} & \mathbf{4 5 - 6 4} & \mathbf{6 5 +} & \text { Total } \\
\hline \text { Republican } & 224 & 340 & 1075 & 561 & \mathbf{2 2 0 0} \\
\hline \text { Democrat } & 184 & 384 & 773 & 459 & \mathbf{1 8 0 0} \\
\hline \text { Total } & \mathbf{4 0 8} & \mathbf{7 2 4} & \mathbf{1 8 4 8} & \mathbf{1 0 2 0} & \mathbf{4 0 0 0} \\
\hline
\end{array}
$$
(a) Are the events "Republican" and "30-44" independent? Justify your answer.
(b) Are the events "Democrat" and "65+" independent? Justify
your answer.
(c) Are the events "17-29" and "45-64" mutually exclusive? Justify your answer.
(d) Are the events "Republican" and "45-64" mutually exclusive? Justify your answer.

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Key Concepts

Contingency Tables

Contingency tables are tabular representations of data that display the frequency distribution of variables. They are commonly used in statistics to analyze the relationship between two categorical variables by providing the counts needed to compute marginal and joint probabilities, which in turn assist in evaluating independence and mutual exclusivity.

Independence

Independence in probability refers to a situation where the occurrence of one event does not affect the probability of another. Mathematically, two events A and B are independent if the joint probability P(A and B) equals the product P(A)P(B). This concept is vital when determining whether two classifications, such as political affiliation and age group, show any dependency in a data set.

Mutual Exclusivity

Mutual exclusivity means that two events cannot occur at the same time; that is, the occurrence of one event precludes the occurrence of the other. If events A and B are mutually exclusive, their joint probability P(A and B) is zero. This concept is fundamental in recognizing situations where classifications or outcomes have no overlap.

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