Suppose that 100 items are drawn from a population of manufactured products and the weight,X,of each item is recorded.Prior experience has shown that the weight has a non-normal probability distribution with $\mu$ = 8 ounces and $\sigma$ = 3 ounces.Which of the following is true about the sampling of $\bar { X }$ ?
A) Its mean is 8 ounces.
B) Its standard error is 0.3 ounces.
C) Its shape is approximately normal.
D) All of these choices are true.

Question

Quizplus · Accepted Answer

A) The mean weight of the population is 8 ounces, so by the properties of the sample mean, the mean of the sampling distribution of the sample mean (i.e., the mean of $\bar{X}$) is also 8 ounces.

B) The standard error of the sample mean can be calculated using the formula $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. In this case, $\sigma$ is given as 3 ounces and the sample size is 100, so the standard error is $\frac{3}{\sqrt{100}} = 0.3$ ounces.

C) The central limit theorem states that, under certain conditions, the sampling distribution of the sample mean will be approximately normal regardless of the shape of the population distribution. One of these conditions is that the sample size is sufficiently large. In this case, the sample size is 100, which is generally considered to be large enough for the central limit theorem to apply.

Therefore, all three choices are true.

Suppose That 100 Items Are Drawn from a Population of Manufactured