Question

question 7 (1 point)
answer the question true or false.

the standard error of the sampling distribution of the sample mean is equal to \\(\sigma\\), the standard deviation of the population.

true
false

question 8 (1 point)
answer the question true or false.

sample statistics are random variables, because different samples can lead to different values of the sample statistics.

true
false

Explanation:

Analyze Question 7

The statement in Question 7 is: "The standard error of the sampling distribution of the sample mean is equal to \(\sigma\), the standard deviation of the population."

Let's recall the formula for the standard error of the sample mean, denoted as \(\sigma_{\bar{x}}\). For a sample of size \(n\) drawn from a population with standard deviation \(\sigma\), the standard error is given by:

$$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$

This standard error is equal to \(\sigma\) only when the sample size \(n = 1\). In general statistical practice, where \(n > 1\), the standard error is strictly less than the population standard deviation \(\sigma\). Therefore, the statement is false.

Analyze Question 8

The statement in Question 8 is: "Sample statistics are random variables, because different samples can lead to different values of the sample statistics."

A sample statistic (such as the sample mean \(\bar{x}\) or sample standard deviation \(s\)) is a numerical summary calculated from a random sample. Because the sample is chosen randomly, its elements vary from one sample to another. Consequently, the calculated statistic varies from sample to sample. This variability is known as sampling variability, and because its value depends on the outcome of a random trial (selecting a sample), a sample statistic is indeed a random variable. Therefore, the statement is true.

Answer:

Question 7

• True
• False (Correct answer)

Question 8

• True (Correct answer)
• False