Question

A simple random sample of size nequals60 is obtained from a population with muequals69 and sigmaequals5. Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why? What is the sampling distribution of x overbar​? Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why? A. 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 overbar become approximately normal as the sample​ size, n, increases. B. Yes because the Central Limit Theorem states that the sampling variability of nonnormal populations will increase as the sample size increases. C. Yes because the Central Limit Theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x overbar ​normal, regardless of the sample​ size, n. D. No because the Central Limit Theorem states that regardless of the shape of the underlying​ population, the sampling distribution of x overbar becomes approximately normal as the sample​ size, n, increases. Your answer is correct. What is the sampling distribution of x overbar​? 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 overbar is skewed left with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing. B. The sampling distribution of x overbar follows​ Student's t-distribution with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing. C. The sampling distribution of x overbar is normal or approximately normal with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing. D. The sampling distribution of x overbar is uniform with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing.

Answer 1

The mean of the population is and the standard deviation is

A sample of size n=60 is drawn from this population.

Here the sample size is greater than 30 and hence the central limit theorem applies. That is as per the central limit theorem, the distribution of sample mean is approximately normal, irrespective of the distribution of the population.

Does the population need to be normally distributed for the sampling distribution of to be approximately normally​ distributed?

ans: D. No because the Central Limit Theorem states that regardless of the shape of the underlying​ population, the sampling distribution of becomes approximately normal as the sample​ size, n, increases.

What is the sampling distribution of ?

has normal distribution with mean and standard deviation ( or called standard error of mean)

ans: C. The sampling distribution of is normal or approximately normal with and