lwfrgin

2021-01-31

To explain:Whether the given statement is correct or not

Faiza Fuller

Given information:
The given statement is,
"if $\stackrel{―}{x}$ is the mean of a large (n > 30) simple random from a population with mean \mu and standard deviation $\sigma$ , then $\stackrel{―}{x}$ is approximately normal with ${\sigma }_{\stackrel{―}{x}}=\frac{\sigma }{\sqrt{n}}$.
The central limit theorem is one of the important concepts of the large sample theory.
According to the central limit theorem, for a sufficiently large sample size, the sampling distributions of the mean tend to be normal distribution, irrespective of the distribution of the population.Generally, a sample size more than 30 is considered a large sample.
The sampling distribution of the sample mean bar x for the large samples follows normal distribution with mean and variance $\frac{{\sigma }^{2}}{n}$ , where mu is the population mean, n is the sample size and ${\sigma }^{2}$ is the population variance, that is,
$\stackrel{―}{x}simN\left(\mu ,\frac{\sigma }{\sqrt{n}}\right)$
$\stackrel{―}{x}simN\left(\mu ,{\sigma }_{\stackrel{―}{x}}\right)$
Thus, the provided statement is true.

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