A simple random sample of size n = 69 is obtained from a population from a popul

Question

A simple random sample of size n = 69 is obtained from a population from a population with mu = 59 and sigma = 7. Does the population need to be normally distributed for the sampling distribution of x bar to be normally distribution? Why? What is the sampling distribution of x bar? Does the population need to be normally distributed for the sampling distribution of x bar to be approximately normally distributed? Why? A. 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. B. Yes because the Central Limit Theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x bar normal, regardless of the sample size, n. C. Yes because the Central Limit Theorem states that the sampling variability of nonnormal populations will increase as the sample size increases. D. No because the Central Limit Theorem states that only if the shape of the underlying population is normal or uniform does the sampling distribution of x bar become approximately normal as the sample size, n, increases. What is the sampling distribution of x bar? Select the correct choice below and fill in the answer boxes within your choice. (Type integers or decimals rounded to three decimal places as needed.) A. The sampling distribution of x bar is uniform with mu^-_x = and sigma^-_x = B. The sampling distribution of x bar follows Student's -distribution with mu^-_x = and sigma^-_x = C. The sampling distribution of x bar is normal or approximately normal with mu^-_x = and sigma^-_x = D. The sampling distribution of x bar is skewed left with mu^-_x = and sigma^-_x =

Explanation / Answer

1. option A is right

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.option C is right

regardless of underlying distribution the sampling becomes normal

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