Problem 19 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

REASONING Will replicating an experiment on many individuals produce data that are more likely to accurately represent a population than performing the experiment only once? Explain.

Step-by-Step Solution

Answer

Yes, replicating an experiment on many individuals is more likely to produce data that accurately represents a population. This is because a larger sample size generally provides a more accurate approximation of the population characteristics, reducing sampling error, and increasing data representativeness.

Step 1: Understand the Principle of Statistical Sampling

In statistics, a sample refers to a subset of a larger population. It provides a snapshot of how the overall population behaves and allows predictions about the population. Therefore, a larger sample size within a population generally tends to provide a more accurate approximation of the population characteristics than a smaller one.

Step 2: Recognize the Impact of Larger Sample Size

Increasing the sample size (in this case, replicating the experiment on many individuals) increases the number of observed outcomes. This, in turn, increases the likelihood that the sample accurately represents the population. The larger the sample size, the smaller the sampling error, thereby providing results that are more likely to reflect the true characteristics of the population.

Step 3: Conclusion

Based on the fundamentals of statistical sampling, replicating an experiment on many individuals would indeed yield data that are more likely to accurately represent a population than performing the experiment only once. This is due to the reduced margin of error and increased representativeness in larger samples.

Key Concepts

Sample Size

One of the crucial elements in statistical sampling is the sample size. This term simply refers to the number of observations or individual data points in a sample. The sample size is key to ensuring that the sample adequately reflects the overall population.

A larger sample size generally translates to more reliable and accurate results. This is because it encompasses a broader representation of the population, reducing the impact of anomalies or outliers.

• Larger sample sizes lead to more consistent results.
• They help minimize the effects of random errors in the data.
• They enhance the credibility of statistical findings.

If you imagine each data point as a piece of a giant puzzle, more pieces (or data points) allow you to see the entire picture more clearly.

Sampling Error

Sampling error is a vital concept to grasp when dealing with statistical sampling. It refers to the difference between the results obtained from a sample and the true characteristics of the entire population.

Any sample will have some level of sampling error because it is only a portion of the whole population. However, understanding this concept helps in designing better experiments and improving statistic accuracy.

• Reduced through increased sample size.
• An inherent part of sampling but controllable.
• Can be minimized with proper sampling techniques.

Think of sampling error as the discrepancy between a zoomed-in focused picture (sample) and a panoramic view (full population).

Population Characteristics

In statistical studies, population characteristics refer to the features or qualities of a group being studied. These traits can vary widely depending on what the research is focused on, such as age, income, preferences, or behaviors. Recognizing and understanding these characteristics is essential when conducting experiments and analyses because they are the ultimate target of predictions and conclusions.

Population characteristics guide researchers in setting parameters for their samples:

• It ensures samples match the research goals.
• Helps in understanding potential variances within the data.
• Aids in designing experiments that can better represent reality.

By accurately capturing these characteristics, we increase the reliability and relevance of our findings, making predictions based on our sample data much more applicable to the entire population.