true or false If x represents a random variable with mean 114 and standard deviation 40, then the standard deviation of the sampling distribution of the means with sample size 100 is 4.

We also know that we select a sample size of n =100 and on this case since the sample size is higher than 30 we can apply the central limit theorem and the distribution for the sample mean would be given by:

[tex] \bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And the standard deviation for the sampling distribution would be:

[tex]\sigma_{\bar X}= \frac{40}{\sqrt{100}}= 4[/tex]

So then the answer is TRUE

Step-by-step explanation:

Let X the random variable of interest and we know that the true mean and deviation for this case are given by:

[tex]\mu = 114, \sigma = 40[/tex]

We also know that we select a sample size of n =100 and on this case since the sample size is higher than 30 we can apply the central limit theorem and the distribution for the sample mean would be given by:

[tex] \bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And the standard deviation for the sampling distribution would be:

[tex]\sigma_{\bar X}= \frac{40}{\sqrt{100}}= 4[/tex]

So then the answer is TRUE