Problem 13 - Biology Concepts and Applications

Question

Fifteen randomly selected students are found to be taller than 6 feet. The researchers concluded that the average height of a student is greater than 6 feet. This is an example of ______________. a. experimental error c. a subjective opinion b. sampling error d. experimental bias

Step-by-Step Solution

Answer

This is an example of sampling error.

Step 1: Understand the Exercise

We are given that 15 randomly selected students are taller than 6 feet. The researchers concluded that the average height of a student is greater than 6 feet based on this sample. We need to identify the type of error or reasoning they used from the given options.

Step 2: Analyze the Conclusion

The conclusion that the average height is greater than 6 feet is drawn from a limited sample of 15 students. This might not accurately reflect the overall student population.

Step 3: Define Sampling Error

Sampling error occurs when the sample does not represent the entire population, leading to incorrect conclusions. This happens when a sample is used to make inferences about the population.

Step 4: Determine the Cause of Misjudgment

Since only a small group of randomly selected students were found to be taller than 6 feet, the researchers may have based their conclusion on a non-representative sample, indicating a potential sampling error.

Key Concepts

Random Sampling

Random sampling is a method used to select a subset of individuals from a larger population. Each member has an equal chance of being chosen.
This technique helps to minimize bias because it does not favor any part of the population.
For example, if you want to study the height of students in a school, random sampling would mean that every student has the same chance of being included in the sample. Random sampling is crucial because:

• It ensures that the sample accurately reflects the entire population.
• It reduces the risk of selection bias.
• It allows for the generalization of results back to the whole population.

However, just because a sample is selected randomly doesn't guarantee it will perfectly represent the population. That's where the concept of a representative sample comes into play.

Representative Sample

A representative sample is a small group that closely mirrors the characteristics of the entire population.
This type of sample can give us a true picture of the population being studied. In order to draw accurate conclusions, researchers strive to use representative samples.
For instance, if researchers are studying student height, a representative sample would include students of various heights, not just those above a certain threshold. The importance of a representative sample includes:

• Providing a more accurate reflection of the population's traits.
• Enabling reliable statistical inference.
• Reducing the sampling error, which occurs when conclusions drawn from a sample do not accurately reflect the population.

Achieving a representative sample can sometimes be challenging, particularly when random sampling results in a group that isn't reflective of the broader population, as shown in the exercise given.

Statistical Inference

Statistical inference involves making conclusions about a larger population based on data collected from a sample. It is a key aspect of statistics that enables researchers to make educated guesses.
The conclusions often involve estimating characteristics like the population mean or variance. Statistical inference techniques include:

• Hypothesis testing, which determines if there is enough evidence to support a particular claim about the population.
• Confidence intervals, which provide a range of values that likely contain the population parameter.

However, inferences can be faulty if the sample is not representative. As seen in the exercise, assuming that the average student height is greater than 6 feet based on a limited sampling may lead to incorrect conclusions. This highlights the importance of using carefully obtained, representative samples to support statistical inference.