Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a poll conducted among likely voters by Zogby International, voters were asked their opinion on the best alternative to oil and coal. The results are as follows: $$ \begin{array}{lcccccc} \hline & & & \text { Fuel } & & \text { Other/ } \\ \text { Source } & \text { Nuclear } & \text { Wind } & \text { cells } & \text { Biofuels } & \text { Solar } & \text { no answer } \\ \hline \text { Respondents, } \% & 14.2 & 16.0 & 3.8 & 24.3 & 27.9 & 13.8 \\ \hline \end{array} $$ What is the probability that a randomly selected participant in the poll mentioned a. Wind or solar energy sources as the best alternative to oil and coal? b. Nuclear or biofuels as the best alternative to oil and coal?

Step-by-Step Solution

Answer

a. The probability that a randomly selected participant mentioned Wind or Solar energy sources as the best alternative to oil and coal is \(43.9\%\). b. The probability that a randomly selected participant mentioned Nuclear or Biofuels energy sources as the best alternative to oil and coal is \(38.5\%\).

Step 1: Locate Wind and Solar values in the table

In the table provided, find the percentage of respondents who chose Wind and Solar energy sources. For Wind, it's 16.0% and for Solar, it's 27.9%.

Step 2: Add the percentages

Now, simply add the percentages of respondents who chose Wind and Solar energy sources to find the probability of a randomly selected participant mentioning either Wind or Solar energy sources as the best alternative to oil and coal. \(P (\text{Wind or Solar}) = 16.0 \% + 27.9 \% = 43.9 \%\) b. Nuclear or biofuels as the best alternative to oil and coal:

Step 1: Locate Nuclear and Biofuels values in the table

In the table provided, find the percentage of respondents who chose Nuclear and Biofuels energy sources. For Nuclear, it's 14.2% and for Biofuels, it's 24.3%.

Step 2: Add the percentages

Now, simply add the percentages of respondents who chose Nuclear and Biofuels energy sources to find the probability of a randomly selected participant mentioning either Nuclear or Biofuels energy sources as the best alternative to oil and coal. \(P (\text{Nuclear or Biofuels}) = 14.2 \% + 24.3 \% = 38.5 \%\) So, a. The probability that a randomly selected participant mentioned Wind or Solar energy sources as the best alternative to oil and coal is \(43.9\%\). b. The probability that a randomly selected participant mentioned Nuclear or Biofuels energy sources as the best alternative to oil and coal is \(38.5\%\).

Key Concepts

Probability Calculation

Understanding probability is crucial when interpreting poll results like those from Zogby International regarding alternative energy preferences. Probability describes the likelihood of a particular outcome and is usually expressed as a percentage or a fraction.

For example, to find the probability that a randomly selected participant in the poll prefers wind or solar energy, you sum the individual probabilities of each outcome since they are mutually exclusive. This is represented mathematically as:
\[P(\text{Wind or Solar}) = P(\text{Wind}) + P(\text{Solar})\]
This calculation assumes that a participant cannot choose both wind and solar, hence the use of addition in our probability calculation. It's important to note that the probabilities must be converted from percentages to decimals if you're performing any mathematical operations. However, in the context of the solution, adding the percentages directly is sufficient because we're interested in the combined probability in percentage form.

To improve on the steps in the solution, we can consider the law of total probability if analyzing more intricate polls where options may overlap or when considering 'none of the above' choices, or the probabilities of choosing several options combined.

Alternative Energy Sources

In the context of the given poll, alternative energy sources refer to options other than fossil fuels, which include oil and coal. These alternatives have gained attention due to their potential for reducing greenhouse gas emissions and mitigating climate change. The main alternatives mentioned are:

• Nuclear energy: Generated from nuclear reactions, providing a large amount of energy with low greenhouse gas emissions once the plant is operational.
• Wind energy: Harnessed from wind via turbines, it's renewable and produces no direct emissions during operation.
• Fuel cells: Devices that convert chemical energy directly into electrical energy, often using hydrogen as a clean fuel.
• Biofuels: Made from organic materials, biofuels are renewable and considered to reduce certain types of emissions.
• Solar energy: Derived from the sun, solar power can be harnessed via panels or thermal systems.

Understanding public opinion on these energy sources helps policymakers and companies to plan for the future energy mix. Each alternative has its own advantages and challenges, factoring into the preferences of the participants in the poll.

Statistical Analysis

Polls like the one conducted by Zogby International are not just inquiries but involve rigorous statistical analysis. The process starts with the design of the poll, including question formulation and determining the sample size that will accurately represent the population.

When the data is collected, statistical techniques are employed to analyze the responses. Descriptive statistics summarize the data – in this case, the percentage of respondents favoring each alternative energy source. Inferential statistics might then be used to generalize the results to a larger population, deducting margins of error or confidence intervals.

In the provided solution, adding percentages directly worked because it was based on a simple probability model. However, noting that real-world data often involves more complexity, like weighing responses or considering sampling errors, would improve the analysis. The accuracy of a poll is determined not only by the data but also by how it is interpreted and analyzed, making statistical expertise key to understanding public opinion captured through polls.