Problem 126 - Applied Statistics and Probability for Engineers

Question

In the 2012 presidential clection, exit polls from the critical state of Ohio provided the following results: $$\begin{array}{lcc}\text { Total } & \text { Obama } & \text { Romney } \\\\\text { No college degree }(60 \%) & 52 \% & 45 \% \\\\\text { Collcge graduate }(40\%) & 47 \% & 51 \%\end{array}$$ What is the probability a randomly selected respondent voted for Obama?

Step-by-Step Solution

Answer

The probability a randomly selected respondent voted for Obama is 50%.

Step 1: Break Down the Information

The exit poll results are broken down into two groups: 'No College Degree' and 'College Graduate'. The proportion of each group is given (60% and 40%, respectively). The support for Obama and Romney in each group is also provided.

Step 2: Calculate Probability in Each Group

Calculate the probability of voting for Obama in each group. For the 'No College Degree' group, it's 52% and for the 'College Graduate' group, it's 47%.

Step 3: Weight Each Probability by Group Size

Multiply the probability of voting for Obama in each group by the proportion of that group within the total population.For 'No College Degree': \(0.60 \times 0.52 = 0.312\)For 'College Graduate': \(0.40 \times 0.47 = 0.188\)

Step 4: Sum the Weighted Probabilities

Add the weighted probabilities from each group to find the total probability that a randomly selected respondent voted for Obama.Total probability for Obama = \(0.312 + 0.188 = 0.500\) or 50%.

Key Concepts

Exit Poll Analysis

Exit Poll Analysis is a method used to predict the outcome of an election. This involves surveying voters immediately after they have left the polling stations. The data collected, which includes demographic information and voting choices, helps analysts understand voting patterns and electoral behavior. By analyzing this information, we can draw insights into how different groups have voted and why. In the 2012 presidential election, the exit polls from Ohio provided crucial information on voter preferences divided by educational attainment. This separation into 'No College Degree' and 'College Graduate' groups allowed for a detailed analysis of how these distinct groups supported the candidates, offering insights into the demographics of Obama and Romney supporters.

Weighted Probability

Weighted Probability is a technique used to consider the impact of different groups on a statistical measure. Instead of treating each subset of data equally, weighted probability assigns different significance to each group based on its size or importance.In the context of the 2012 Ohio exit polls, the concept is applied by giving weight to the voting probabilities of the 'No College Degree' group and the 'College Graduate' group. Since 60% of the respondents were from the 'No College Degree' group, this group had a larger impact on calculating the overall probability that a randomly selected respondent voted for Obama.Calculation involved multiplying the probability within each group by its respective size within the total population:

• For 'No College Degree': the weighted probability is \(0.60 \times 0.52 = 0.312\).
• For 'College Graduate': it is \(0.40 \times 0.47 = 0.188\).

Weighted probability thus ensures a more accurate reflection of the overall voting tendency among diverse groups.

Statistical Groups

Statistical Groups refer to categorized subsets within a larger population used in data analysis. Grouping data allows analysts to observe trends and differences between categories, enhancing the understanding of complex datasets. In the 2012 exit poll, respondents were divided into two main statistical groups based on education: 'No College Degree' and 'College Graduate'. This classification helps reveal how educational attainment correlates with voting preferences. Such grouping is vital since it can highlight significant variations in behavior or opinions between demographics. For instance, in the Ohio polls, a greater percentage of those without a college degree voted for Obama, whereas Romney had stronger support from college graduates. Understanding these distinct patterns helps in crafting more targeted political strategies and policies.

Voting Patterns

Voting Patterns indicate the tendency of specific demographic segments to vote in particular ways, informed by characteristics like age, education, economic status, and more. In the context of the 2012 Ohio exit polls, significant voting patterns were observed. Voters with no college degree tended to support Obama more than Romney, while college graduates preferred Romney. Such patterns are crucial for political analysis as they offer insights into the electorate's priorities and what motivates their decisions. Identifying these trends can help candidates tailor their campaigns. Political parties often analyze these patterns to understand the electorate better and devise strategies that resonate with major voter blocks. Recognizing these trends in segments like education makes it possible to focus on issues that matter to these groups, thus potentially swaying future voting behavior in favor.