Given information:
The given statement is,
"if \overline{x} is the mean of a large (n > 30) simple random from a population with mean \mu and standard deviation \sigma , then \overline{x} is approximately normal with \sigma_{\overline{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,
\(\overline{x}sim N(\mu,\frac{\sigma}{\sqrt{n}}\)
\(\overline{x}sim N(\mu,\sigma_{\overline{x}}\)
Thus, the provided statement is true.
The given statement is,
"if \overline{x} is the mean of a large (n > 30) simple random from a population with mean \mu and standard deviation \sigma , then \overline{x} is approximately normal with \sigma_{\overline{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,
\(\overline{x}sim N(\mu,\frac{\sigma}{\sqrt{n}}\)
\(\overline{x}sim N(\mu,\sigma_{\overline{x}}\)
Thus, the provided statement is true.
