Question

Approximately normally distributed? Why? What is the sampling distribution of xˉ

? Does the population need to be normally distributed for the sampling distribution of xˉ
to be approximately normally distributed? Why? A. Yes. The central limit theorem states that the sampling variability of nonnormal populations will increase as the sample increases. increases. What is the sampling distribution of xˉ
? 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 xˉ
is skewed left with μ Xˉ−​ = and σ Xˉ− = B. The sampling distribution of xˉ is approximately normal with μ xˉ − =and σ xˉ− = C. The shape of the sampling distribution of xˉ is unknown with μ Xˉ − =and σ Xˉ−​ = D. The sampling distribution of xˉ is uniform with μ xˉ −​ =and σ xˉ −​ =

Answer 1

The sampling distribution of the mean becomes approximately normal due to the Central Limit Theorem, which applies regardless of the population distribution as the sample size becomes large. Option B is correct: the sampling distribution is approximately normal with a mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.

Step-by-step explanation:

The sampling distribution of the mean, denoted as Xbar , approaches an approximately normal distribution as the sample size increases, regardless of the shape of the population distribution. This is described by the Central Limit Theorem, which specifies that this distribution will have a mean (μXbar ) equal to the population mean (μ) and a standard deviation (σXbar ), known as the standard error, equal to the population standard deviation (σ) divided by the square root of the sample size (n).

Therefore, the correct choice regarding the sampling distribution of Xbar is: B. The sampling distribution of X bar is approximately normal with μ X bar = μ and σXbar = σ/√n. The population does not need to be normally distributed for the sampling distribution of X bar to be approximately normally distributed, thanks to the Central Limit Theorem. As the sample size increases, the sampling distribution will increasingly resemble a normal distribution.