Problem 10 - Prealgebra

Question

Identify each sample as biased or unbiased and describe its type. Explain your reasoning. To determine whether a new university library would be useful, all students whose student ID number ends in 2 are surveyed.

Step-by-Step Solution

Answer

The sample is biased and systematic because it selects students based on ID number, not randomly.

Step 1: Identify the Type of Sampling

The scenario describes a method for selecting a sample to determine if a new university library would be useful. All students whose student ID number ends in 2 are chosen to be surveyed. This selection is likely systematic as it relies on a fixed property of the students' IDs.

Step 2: Determine if the Sample is Biased or Unbiased

A sample is considered unbiased if every individual has an equal chance of being selected. In this case, only students with IDs ending in 2 are being selected, which means the sample is not random and does not give each student an equal chance of being chosen. This could lead to a biased sample, as the selection might not represent the entire student body fairly.

Step 3: Conclude the Bias Nature and Type

Since the sample is obtained through non-random selection based on ID numbers, it is systematic but biased. Systematic samples can become biased if the system used (in this case, student ID numbers) excludes potential variations that might exist in the broader population of all students.

Key Concepts

Systematic Sampling

Systematic sampling is a technique used in statistics for selecting a sample from a larger population. It involves selecting members of the population at regular intervals, often based on a pattern or rule. Let's say you have a list of 1000 students and you want to select every 10th person on the list. If you start from the 2nd student, the sample will include students with IDs ending in 2, 12, 22, and so on.

There are benefits to systematic sampling:

• It is straightforward and easy to implement.
• It's efficient for very large populations where individual selection might be impractical.

However, systematic sampling can lead to issues if the pattern interacts with another pattern in the population. For example, if student ID numbers are assigned based on certain criteria, selecting every 10th ID might introduce bias. This is why it's crucial to understand the structure of the population before choosing systematic sampling.

Biased Samples

A biased sample is one that does not accurately represent the population from which it was drawn. When a sample is biased, it tends to favor certain outcomes over others, which can lead to inaccurate conclusions. Bias can occur in several ways:

• Selection bias: When every member of the population does not have an equal chance of being selected.
• Response bias: When the respondents have a tendency to answer questions in a particular way, often influenced by the survey method or environment.

In the given scenario, only students with ID numbers ending in 2 are surveyed, which means not every student has an equal chance of selection. This results in selection bias, as certain groups within the broader student population are excluded. This can affect the results, making them unrepresentative of the opinions or behaviors of all students.

Unbiased Samples

An unbiased sample is designed to fairly represent the entire population from which it is drawn, ensuring that every individual has an equal chance of selection. This is crucial for drawing accurate conclusions from the sample data. Characteristics of unbiased samples include:

• Random selection: Every member of the population has an equal probability of being chosen. This can often be achieved through techniques like simple random sampling or stratified sampling.
• Representativeness: The sample mirrors the diversity and characteristics of the population, reducing the potential influence of outliers or fringe cases.

For unbiased sampling, it's essential to avoid any patterns or criteria that could skew the selection process. By ensuring every student's opinion is equally likely to be heard, we increase the validity of the survey results and the conclusions that follow from them.