Question

Suppose that the mean weight of a certain type of pig is $280pounds$with a standard deviation of$80pounds$. The weight distribution of pigs tends to be somewhat skewed to the right. A random sample of $100$pigs is taken. Which of the following statements about

the sampling distribution of the sample mean weight $\overline{x}$is true?

(a) It will be Normally distributed with a mean of $280pounds$and a standard deviation of $80pounds.$

(b) It will be Normally distributed with a mean of $280pounds$and a standard deviation of $8pounds.$

(c) It will be approximately Normally distributed with a mean of $280pounds$and a standard deviation of $80pounds.$

(d) It will be approximately Normally distributed with a mean of $280pounds$and a standard deviation of $8pounds.$

(e) There is not enough information to determine the mean and standard deviation of the sampling distribution.

Short answer

The answer is $8$so, option (d) is correct.

Step-by-step solution

Given information

We have been given that the distribution will be approximately normally distributed.

Now we choose correct option for the statement about the sampling distribution of the sample mean weight$\overline{x}$is true.

Explanation

We cannot say that the distribution is exactly normally distributed, but since $n>30$we can say that by the central limit theorem, the distribution is approximately normal.

$\mu =280$

The mean for a sample is the same as the population mean.

$\sigma =8$

$\sigma =\frac{80}{\sqrt{100}}=8$