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Let Y₁, Y2,..., Yn is a random sample from a distribution from a distribution with finite mean and variance o2, then sampling distribution of sample means follows always a normal distribution always a normal distribution for any sample size approximately normal distribution for larger sample size O approximately normal distribution for any sample size

Let Y₁, Y2,..., Yn is a random sample from a distribution from a distribution with finite mean and variance o2, then sampling distribution of sample means follows always a normal distribution always a normal distribution for any sample size approximately normal distribution for larger sample size O approximately normal distribution for any sample size

MATLAB: An Introduction with Applications
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 Distributions of Sample Means** Given \( Y_1, Y_2, \ldots, Y_n \) as a random sample from a distribution with a finite mean \(\mu\) and variance \(\sigma^2\), the sampling distribution of the sample means follows: - \(\circ\) always a normal distribution - \(\circ\) always a normal distribution for any sample size - \(\circ\) approximately normal distribution for larger sample size - \(\circ\) approximately normal distribution for any sample size ### Explanation: When dealing with the distribution of sample means, the Central Limit Theorem (CLT) plays a crucial role. The CLT states that the sampling distribution of the sample mean will tend to be normal, or nearly normal, if the sample size is sufficiently large, regardless of the shape of the population distribution. This is why the correct answer to this multiple-choice question is usually "approximately normal distribution for larger sample size." The other options are incorrect because: 1. The sample mean is not "always a normal distribution." The original distribution plays a role, especially with small sample sizes. 2. It is not "always a normal distribution for any sample size." Small sample sizes may not yield a normal distribution, especially if the original population distribution is not normal. 3. The option "approximately normal distribution for larger sample size" aligns with the Central Limit Theorem. 4. "Approximately normal distribution for any sample size" is not generally correct, as approximation improves with larger sample sizes. Understanding this principle is fundamental in statistics, enabling more accurate inferences about population parameters from sample statistics.
Transcribed Image Text:**Understanding Sampling Distributions of Sample Means** Given \( Y_1, Y_2, \ldots, Y_n \) as a random sample from a distribution with a finite mean \(\mu\) and variance \(\sigma^2\), the sampling distribution of the sample means follows: - \(\circ\) always a normal distribution - \(\circ\) always a normal distribution for any sample size - \(\circ\) approximately normal distribution for larger sample size - \(\circ\) approximately normal distribution for any sample size ### Explanation: When dealing with the distribution of sample means, the Central Limit Theorem (CLT) plays a crucial role. The CLT states that the sampling distribution of the sample mean will tend to be normal, or nearly normal, if the sample size is sufficiently large, regardless of the shape of the population distribution. This is why the correct answer to this multiple-choice question is usually "approximately normal distribution for larger sample size." The other options are incorrect because: 1. The sample mean is not "always a normal distribution." The original distribution plays a role, especially with small sample sizes. 2. It is not "always a normal distribution for any sample size." Small sample sizes may not yield a normal distribution, especially if the original population distribution is not normal. 3. The option "approximately normal distribution for larger sample size" aligns with the Central Limit Theorem. 4. "Approximately normal distribution for any sample size" is not generally correct, as approximation improves with larger sample sizes. Understanding this principle is fundamental in statistics, enabling more accurate inferences about population parameters from sample statistics.
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