Question

Suppose that the mean weight of a certain breed of pig is $280$pounds with a standard deviation of $80$pounds. The distribution of weight for these 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 $280$pounds and a standard deviation of $80$pounds.

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

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

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

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

Short answer

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

Step-by-step solution

Concept introduction

In quantitative tests, segmentation is the technique of selecting a predefined dataset from a huge population. Vary based on the type of assessment being undertaken, the measures taken to recruit from a general community may include simple chance picking or multi - stage sampling.

Explanation

Assume that the mean weight of a specific trait of pig is $280$pounds, with a $80$pound confidence interval. The weight distribution for these pigs is somewhat biased to the right.

As a result, the incidence will be essentially normal. We cannot assert that the probability is perfectly normal, but because $n>30$, we may say that it is normally distributed according to the central limit theorem. As a result,

$$\begin{gathered} \sigma =\frac{80}{\sqrt{100}} \\ =8 \\ \mu =280 \end{gathered}$$

Option (d) is correct