13. Household income in the United States is strongly skewed to the right. The
current presidential administration claims that the mean household income is
greater than it has ever been in the past. An independent contractor will obtain
a random sample of 100 households in the United States and will calculate the
mean household income. Which of the following statements is true?
(A) The sampling distribution of the sample mean household income is approximately
normal because the sample size of 100 is greater than 30.
(B) The distribution of household income for the sample is approximately normal
because the sample size of 100 is greater than 30.
(C) The sampling distribution of the sample mean household income is strongly
skewed to the right because the population standard deviation is unknown.
(D) The distribution of household income is strongly skewed to the left because the
population sample size of 100 is greater than 30.
(E) The sampling distribution of the sample mean household income is strongly skewed
to the right because the population distribution is strongly skewed to the right.

We have a random variable, that is, household income in the United States.
The question tells us that this variable is strongly skewed to the right. It means that its tail is on the right. The distribution is not symmetrical.
We are going to obtain a random sample from this. In this case, this sample will be of 100 household incomes.

Imagine we have this random sample and calculate its mean: we sum all the household incomes and then divide the result by 100. We have the mean for this sample. If we repeat this for different samples, we obtain different means and we can construct a sampling distribution of the sample means.

In this sampling distribution, we have different probabilities for each sample mean. Some are more probable than others. The interesting part of this is as follows: no matter the distribution these sample means to come from, the sampling distribution of the sample means is approximately normal for samples of size equal or greater than 30 observations. The larger the sample size (n), the more normal will be the shape of the sampling distribution. This result is known as the central limit theorem.

As the number of observations increases, this distribution is more similar to a normal distribution, with a mean that equals the population mean, [tex] \\ \mu[/tex], and a standard deviation equal to [tex] \\ \frac{\sigma}{\sqrt{n}}[/tex].

It can be described mathematically as follows:

[tex] \\ \overline{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

From this explanation, we can deduce that the option (A) is true because it tells us that

"The sampling distribution of the sample mean household income is approximately normal because the sample size of 100 is greater than 30"

That is true because the statement is telling us that the sampling distribution of the sample means, [tex] \\ \overline{X}[/tex], is approximately normal for a sample size greater than 30 observations (in this case, 100 observations).

And with this information, we can make inferences about the true mean of the population, in this case, the household incomes in the United States.

Therefore, if we analyze the next statements, we can conclude that:

Option (B) is false. We already know that the distribution for the random variable household income (we can say it is represented by X) is skewed to the right, and, according to the question, we do not have information about the distribution this variable, X, comes from.
Option (C) is false. The sampling distribution of the sample means will be approximately normal (symmetrical, not skewed) for samples with 30 or more observations, and we have 100 observations here, and the unknown standard deviation does not cause this skewness or lack of symmetry.

Option (D) is false. As we said before, the distribution of household income (X) is skewed to the right (not to the left) according to the question.
Option (E) is false. The sampling distribution of the sample means, [tex] \\ \overline{X}[/tex], is approximately normal (symmetrical) for samples of size equal to or greater than 30 observations, and in this case, we have 100 observations for this sample.