Problem 10 - Algebra 2

Question

Determine whether each situation would produce a random sample. Write yes or no and explain your answer. asking every twentieth person on a list of registered voters to determine which political candidate is favored

Step-by-Step Solution

Answer

No, because every individual does not have an equal chance of being selected.

Step 1: Understand Random Sampling

A random sample is one in which each member of the population has an equal chance of being selected. This helps ensure that the sample accurately reflects the population as a whole.

Step 2: Describe the Sampling Method

In this scenario, every twentieth person on a list of registered voters is selected to participate in the survey.

Step 3: Analyze Equal Chance in Sampling

Determine if every voter has an equal chance of being selected. In systematic sampling, like this scenario, once the starting point is established, not every voter has an equal chance because only those in positions that are multiples of twenty will be chosen.

Step 4: Determine If the Sample is Random

Since not every individual in the population has an equal probability of selection due to the systematic nature of the sampling (selecting every twentieth person), this is not a truly random sample.

Step 5: Formulate Conclusion on Randomness

The situation does not produce a random sample because the systematic selection does not provide equal chances for all individuals to be included.

Key Concepts

systematic sampling

Systematic sampling is a structured method of selecting a sample from a larger population. Instead of picking randomly, systematic sampling involves selecting elements at regular intervals. For example, every 20th person on a list might be chosen. This method is straightforward, especially if dealing with a list or sequence.

• First, you establish a starting point. This point could be randomly determined to add a layer of randomness to the sample.
• Then, consistently choose participants at fixed intervals, like every 5th or 10th individual.

Systematic sampling is beneficial when working with a very large population because it is less complicated and time-consuming compared to entirely random sampling. However, it's important to be cautious with periodic patterns within the list that might affect representation.

equal chance

The concept of equal chance is crucial in understanding the effectiveness of any sampling method. When each member of a population has an equal probability of being selected for the sample, it is said to have equal chance. This is a key characteristic of random sampling. However, with systematic sampling, equal chance is not guaranteed. After the starting point is determined, only the elements at regular intervals from that point are chosen.

• In our example of choosing every 20th voter, only individuals at those specific positions (20th, 40th, 60th, etc.) have a chance of selection.
• Those not at these positions, no matter what their characteristics, have zero chance of selection.

Thus, equal chance might be compromised, which can affect the sample's ability to truly represent the entire population.

sampling methods

Understanding different sampling methods helps in choosing the right approach for a study. Some of the popular sampling methods are:

• Random Sampling: Every member has an equal chance of selection.
• Systematic Sampling: Selects members at regular intervals from a randomly chosen starting point.
• Stratified Sampling: Divides the population into subgroups (strata) and samples each subgroup.
• Cluster Sampling: Divides population into clusters, then randomly selects entire clusters.

Each method has its pros and cons. For instance, random sampling is ideal for unbiased representation, but it can be resource-intensive. Systematic sampling is easier to implement but can miss variations within periodic lists. Careful consideration of the population and resources is vital when choosing a method.

population representation

Population representation is key to ensuring that conclusions drawn from a sample reflect the broader group. The goal of sampling is to obtain a mini-representation of the entire population, which provides insights that can be generalized back to the whole. In accurately representing a population:

• The sample needs to capture the diversity of the population, including different characteristics and behaviors.
• Proper sampling methods help avoid biases that could skew results.

Failing to achieve proper representation can lead to partial or misleading outcomes. For example, in the political survey scenario with systematic sampling, if voter opinions vary by position on the list, the sample may not truly reflect the views of the entire registered population.