Problem 34 - Algebra 2

Question

Before an election, a company conducts a telephone survey of likely voters. Based on their survey data, the polling company states that an amendment to the state constitution is supported by 59% of the state’s residents and that 41% of the state’s residents do not approve of the amendment. According to the company, the results of their survey have a margin of error of 3%. Write and solve an equation describing the maximum and minimum percent of the state’s residents that support the amendment.

Step-by-Step Solution

Answer

The support ranges from 56% to 62%.

Step 1: Understand the Margin of Error

The survey states a margin of error of 3%. This margin of error affects the reported percentage of 59%, indicating the possible range for the true percentage of residents supporting the amendment.

Step 2: Determine the Maximum Support Percentage

To find the maximum possible percentage of residents who support the amendment, add the margin of error to the reported support percentage: \( 59\% + 3\% = 62\% \).

Step 3: Determine the Minimum Support Percentage

To find the minimum possible percentage of residents who support the amendment, subtract the margin of error from the reported support percentage: \( 59\% - 3\% = 56\% \).

Step 4: Write the Equation

The equation representing the possible support range given the margin of error is:\[ 56\% \leq \text{Support} \leq 62\% \].

Key Concepts

Understanding Telephone Surveys

Telephone surveys are a method many companies and organizations use to gather opinions from a group of people. In this method, surveyors contact selected participants via phone to ask questions, such as who they plan to vote for in an election or their stance on a certain topic. This approach allows researchers to quickly reach a diverse audience and collect data efficiently.
A telephone survey is typically less expensive and faster to conduct compared to face-to-face interviews or mail surveys. It also encourages swift responses, as people are more likely to answer immediate questions over the phone rather than taking the time to fill out a written survey at their convenience.
One must consider that telephone surveys may be subject to biases. Not every individual has access to a phone or may be willing to participate, which can affect the representativeness of the survey results. Understanding these limits is crucial to interpreting the gathered data accurately.

The Role of Percentages in Data Interpretation

Percentages play a crucial role in the interpretation of survey data, as they convert the raw data into a form that is easy to understand and compare. When you read a percentage, it represents a part of a whole. For example, in the survey, 59% indicates that 59 people out of every 100 support the amendment.
Using percentages allows for readily comprehensible comparisons between different groups or categories. It also helps in visualizing data when you want to understand the prevalence or magnitude of a response among a population.
In our case, the percentages from the telephone survey clarify the level of support for a state amendment amongst the population. Calculating these figures correctly is essential for deriving meaningful insights from the survey data.

Calculating Maximum and Minimum from a Margin of Error

The concept of maximum and minimum calculations is necessary when a margin of error is involved. A margin of error provides a range around the survey's reported percentage, acknowledging that the percentage might not be exact. In the problem, with 59% of residents showing support, and a margin of error of 3%, the true support could possibly be higher or lower.
To find the maximum potential support, you add the margin of error to the reported percentage. Thus, the maximum support is calculated as follows:

• 59% + 3% = 62%

To determine the minimum potential support, you subtract the margin of error from the reported percentage:

• 59% - 3% = 56%

Thus, we can conclude that the true support for the amendment likely falls between 56% and 62%. Understanding this concept allows one to appreciate how the sample size and error margins can influence confidence in survey data.