Problem 9 - Prealgebra

Question

Identify each sample as biased or unbiased and describe its type. Explain your reasoning. To determine whether a candidate for governor is popular with the voters, \(30 \%\) of citizens in each of the 254 counties are surveyed.

Step-by-Step Solution

Answer

The sample is unbiased and stratified.

Step 1: Understanding the Population and Sample

The population in this context includes all eligible voters across the 254 counties. The sample includes 30% of citizens from every county.

Step 2: Determine If the Sample Is Proportional

Evaluating the design of the sample, it is proportional because exactly 30% of citizens are surveyed in each of the 254 counties, ensuring each county contributes equally by percentage.

Step 3: Identify Bias or Unbias

Since every county is sampled equally by percentage and no group is excluded, this sample is an unbiased representation of the population.

Step 4: Conclusion and Type

The sample is unbiased and stratified, as it includes equal proportions of individuals from different strata (counties) within the population.

Key Concepts

Biased vs Unbiased Samples

When we talk about samples in research, we often distinguish between biased and unbiased samples. An unbiased sample is one that accurately reflects the population without any systematic exclusion of certain groups. This ensures that the findings from the sample can be generalized to the entire population. Conversely, a biased sample fails to represent the population fairly and can lead to misleading results.

In the given example, a sample where 30% of citizens from each county are surveyed is unbiased. This is because every county contributes equally by percentage. The survey method does not exclude any group, thus ensuring no part of the population is overrepresented or underrepresented. Maintaining such balance is crucial for the validity of the research findings.

Stratified Sampling

Stratified sampling is a technique that involves dividing a population into smaller sub-groups, known as strata, which share similar characteristics. Then, samples are taken from each stratum. The main aim is to ensure that each subgroup is adequately represented, thus improving the accuracy of the sample.

In the example provided, stratified sampling is used by surveying 30% of citizens in each of the 254 counties. Each county acts as a stratum in this approach. By ensuring each stratum contributes equally, one can assume that the sample reflects the diversity of opinions present in the entire population. This method helps avoid bias that might occur if only one particular region or type of demographic were surveyed.

Survey Design

A well-thought-out survey design is crucial for obtaining data that accurately reflects a population. The design involves determining who to survey, how many individuals, and how they will be selected. Well-designed surveys take into account possible biases and aim to eliminate them, enhancing the reliability of the collected data.

In our situation, the survey was designed to gather data by sampling 30% of the population from each county. This design step ensures that the survey reaches diverse segments of the population equally, making the survey results more reliable and valid. Thoughtful planning in survey design like this helps in drawing meaningful conclusions and supports effective decision-making.

Population and Sample Analysis

Understanding the relationship between a population and a sample is fundamental in statistical studies. The population includes all individuals or items of interest, while a sample is a subset of the population that is analyzed to infer insights about the whole group.

In the exercise, the population is all voters across 254 counties, while the sample includes 30% from each county. By doing this, researchers can study the sample to gain insights into the opinions and preferences of voters without surveying every individual. This approach allows for efficient use of resources and time while still achieving a comprehensive analysis of the population.