Question

The sampling distribution of the sample mean often has approximately a normal distribution. As the sample size increases, the sampling distribution of the sample mean has a more bell-shaped appearance. For relatively large sample sizes, the sampling distribution is bell shaped even if the population is highly discrete or highly skewed. This is known as the

Answer 1

Answer: The Central Limit Theorem

Explanation:

The description above explains the central limit theorem which states that given a population having mean U and standard deviation S, and drawing different large sample sizes from the population with replacement, then the distribution of the sample means will be normally distributed approximately and that this condition will be the case whether the distribution is skewed or normal.This hold if the sample size n is at least 30.

Answer 2

Answer:

For this case the best answer would be the "Central Limit theorem" or CLT since we are assuming that we have a large sample size (n>30) so then all the conditions are satisfied to assume a normal distribution for the sample mean $\bar X$

$\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})$

Explanation:

Previous concepts

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

From the central limit theorem we know that the distribution for the sample mean $\bar X$ is given by:

$\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})$

Solution to the problem

For this case the best answer would be the "Central Limit theorem" or CLT since we are assuming that we have a large sample size (n>30) so then all the conditions are satisfied to assume a normal distribution for the sample mean $\bar X$