Question

A simple random sample of size n=69 is obtained from a population that is skewed left with u=48 and o=6. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally​ distributed? Why? What is the sampling distribution of x​?

Answer 1

Solution

1)

No because the Central Limit Theorem states that regardless of the shape of the underlying population, the sampling distributed of x bar becomes approximately normal as the sample size, n, increases.

In probability theory, the central limit theorem (CLT) establishes that, for the most commonly studied scenarios, when independent random variables are added, their sum tends toward a normal distribution (commonly known as a bell curve) even if the original variables themselves are not normally distributed. In more precise terms, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined (finite) expected value and finite variance, will be approximately normally distributed, regardless of the underlying distribution

2)

The sampling distribution of x bar is normal or approximately normal with mu^-_x = 48 and sigma^-_x = 0.722