Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

An opinion poll was conducted among a group of registered voters in a certain state concerning a proposition aimed at limiting state and local taxes. Results of the poll indicated that \(35 \%\) of the voters favored the proposition, \(32 \%\) were against it, and the remaining group were undecided. If the results of the poll are assumed to be representative of the opinions of the state's electorate, what is the probability that a registered voter selected at random from the electorate a. Favors the proposition? b. Is undecided about the proposition?

Step-by-Step Solution

Answer

The probability that a randomly selected voter favors the proposition is \(0.35\), and the probability that a randomly selected voter is undecided about the proposition is \(0.33\).

Step 1: a. Probability of Favoring the Proposition

To find the probability that a randomly selected voter favors the proposition, we can use the percentage given in the problem. Since \(35 \%\) of the voters favor the proposition, the probability that a randomly selected voter favors the proposition is \(\frac{35}{100}\).

Step 2: a. Result

The probability that a randomly selected voter favors the proposition is \(\frac{35}{100}\) or \(0.35\).

Step 3: b. Probability of Being Undecided About the Proposition

To find the probability that a randomly selected voter is undecided about the proposition, we need the percentage of undecided voters. This can be calculated by taking the total percentage of voters (\(100 \%\)) and subtracting the percentage of voters in favor of the proposition (\(35 \%\)) and the percentage of voters against the proposition (\(32 \%\)). This gives us the percentage of undecided voters: \(100 \% - 35 \% - 32 \% = 33 \%\). Therefore, the probability that a randomly selected voter is undecided about the proposition is \(\frac{33}{100}\).

Step 4: b. Result

The probability that a randomly selected voter is undecided about the proposition is \(\frac{33}{100}\) or \(0.33\).

Key Concepts

Opinion Poll Analysis

Understanding opinion poll results is crucial for gauging public sentiment on various issues. In the exercise provided, we delve into an opinion poll regarding a tax-related proposition. The results of this poll offer a snapshot of how the electorate in a particular state feels about limiting state and local taxes.

To analyze these results effectively, it's important to recognize that the poll categorizes voters into three groups: those in favor, those against, and those undecided. By presenting the data into these clear segments, we can assess not only the majority opinion but also the proportion of the electorate that might be swayed by campaigns from either side.

It's essential to note that these figures are representative of the opinions at the time of the poll and can fluctuate based on various factors, including news events, advertising campaigns, and grassroots efforts. Additionally, the margin of error, which is not provided in the exercise but is inherent to all polls, can affect the accuracy of these numbers.

Probability Calculations

Probability calculations are fundamental in interpreting poll results. These calculations tell us the chance of a specific outcome when selecting at random from the population.

When the exercise gives a percentage, like the mentioned 35% of voters favoring the proposition, it translates directly into a probability by converting the percentage into a decimal. Here, a probability of \(0.35\) implies that if we were to randomly pick a voter, there's a 35% chance that they would favor the proposition.

Mathematically, we express this by dividing the percentage by 100, thus \(35\% = \frac{35}{100} = 0.35\). Similarly, for undecided voters, we subtract the percentages of the other two groups from 100%, yielding \(33\%\) or a \(0.33\) probability of a voter being undecided. These calculations are not just number crunching; they offer a way to predict voter inclinations across the entire electorate.

Voter Behavior Statistics

The statistics generated from voter behavior are invaluable in various aspects of political strategizing and forecasting. From the opinion poll presented, the statistics suggest where the electorate stands regarding the tax proposition.

However, beyond mere percentages, voter statistics can also inform us about the potential volatility of the vote. For instance, the 33% of undecided voters represent a significant portion of the population that could be influenced by future campaigning or information releases.

Detailed analysis of statistics, like cross-tabulations that consider demographics, can unveil more intricate patterns in voting behavior, such as tendencies among different age groups or socioeconomic statuses. In light of these details, political strategists often craft targeted messaging in hopes of appealing to these specific subgroups to turn the tides in their favor.