Thank you for visiting According to the central limit theorem the mean of the sampling distribution the mean of the population A is always equal to B is slightly. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
The central limit theorem (CLT) is a fundamental principle in statistics and probability theory, especially relevant when dealing with sampling distributions. According to the central limit theorem, when you take a sufficiently large sample size from a population with a finite level of variance, the mean of the sampling distribution of the sample means will be approximately equal to the mean of the population. Therefore, the correct answer is:
a. is always equal to
Here's a step-by-step breakdown of why this is true:
Central Limit Theorem Basics: The central limit theorem states that regardless of the original distribution of the population (whether normal or not), the distribution of the sample means will tend to be normal (bell-shaped) if the sample size is large enough.
Example of Convergence: For example, if a population mean ([tex]\mu[/tex]) is 50, then with repeated sampling and calculating the mean of those samples, the mean of the sample means will converge to 50.
Importance of Sample Size: Typically, a sample size of 30 or more is considered sufficient for the CLT to hold, which leads to the sample mean ([tex]\bar{x}[/tex]) closely approximating the population mean ([tex]\mu[/tex]).
Application in Real-World Scenarios: This principle is crucial because it justifies the use of sample statistics to estimate population parameters, which is heavily relied upon in fields such as economics, biology, and social sciences.
In conclusion, the mean of the sampling distribution is always equal to the mean of the population, provided samples are large enough.
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