The variance of the sampling distribution of the sample average is smaller than the variance of the underlying population. True O False
The variance of the sampling distribution of the sample average is smaller than the variance of the underlying population. True O False
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Understanding Sampling Distribution and Variance
In statistics, the concept of sampling distribution plays a crucial role in making inferences about a population from a sample. One key aspect is understanding the variance of the sampling distribution compared to the variance of the underlying population.
**Statement:**
"The variance of the sampling distribution of the sample average is smaller than the variance of the underlying population."
**Options:**
- True
- False
This statement is true when considering large enough samples. The variance of the sampling distribution of the sample mean is equal to the variance of the population divided by the sample size, denoted as:
\[ \text{Variance of sampling distribution} = \frac{\sigma^2}{n} \]
where:
- \(\sigma^2\) is the population variance
- \(n\) is the sample size
Thus, as the sample size \(n\) increases, the variance of the sampling distribution decreases, making the sample mean a more precise estimate of the population mean. This relationship is a fundamental principle of the Central Limit Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff21aecd2-9b55-44fe-9108-01ab539cadf4%2F7f9bfd8b-7d7d-4039-b59b-e56cf8a27478%2F0vl9bet_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Sampling Distribution and Variance
In statistics, the concept of sampling distribution plays a crucial role in making inferences about a population from a sample. One key aspect is understanding the variance of the sampling distribution compared to the variance of the underlying population.
**Statement:**
"The variance of the sampling distribution of the sample average is smaller than the variance of the underlying population."
**Options:**
- True
- False
This statement is true when considering large enough samples. The variance of the sampling distribution of the sample mean is equal to the variance of the population divided by the sample size, denoted as:
\[ \text{Variance of sampling distribution} = \frac{\sigma^2}{n} \]
where:
- \(\sigma^2\) is the population variance
- \(n\) is the sample size
Thus, as the sample size \(n\) increases, the variance of the sampling distribution decreases, making the sample mean a more precise estimate of the population mean. This relationship is a fundamental principle of the Central Limit Theorem.
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