Question

which of the following statements about the sampling distribution of the sample mean is incorrect?

the standard deviation of the sampling distribution is \\(\sigma\\).
the sampling distribution is generated by repeatedly taking samples of size \\(n\\) and computing the sample means.
the mean of the sampling distribution is \\(\mu\\).
the sampling distribution is approximately normal whenever the sample size is sufficiently large (\\(n \ge 30\\)).

question 12 (1 point)
solve the problem.

a random sample of \\(n = 100\\) measurements is drawn from a binomial population with probability of success .21. give the mean and the standard deviation of the sampling distribution of the sample proportion, \\(\hat{p}\\).

.21; .02
.21; .407
.21; .041
.21; .008

Explanation:

Analyze the first question

Using the Sampling Distribution of the Sample Mean knowledge point

We evaluate the statements regarding the sampling distribution of the sample mean \(\bar{x}\):

• The mean of the sampling distribution is \(\mu_{\bar{x}} = \mu\). This is correct.
• The sampling distribution is generated by repeatedly taking samples of size \(n\) and computing their means. This is correct.
• The Central Limit Theorem states that the sampling distribution is approximately normal if \(n \ge 30\). This is correct.
• The standard deviation of the sampling distribution (standard error) is \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), not \(\sigma\). Thus, the first statement is incorrect.

Calculate the mean of the sample proportion

Using the Sampling Distribution of the Sample Proportion knowledge point

For a binomial population with success probability \(p = 0.21\) and sample size \(n = 100\):

• The mean of the sampling distribution of the sample proportion \(\hat{p}\) is:

$$ \mu_{\hat{p}} = p = 0.21 $$

Calculate the standard deviation of the sample proportion

Using the Sampling Distribution of the Sample Proportion knowledge point

The standard deviation (standard error) of the sample proportion \(\hat{p}\) is:

$$ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.21 \times 0.79}{100}} = \sqrt{\frac{0.1659}{100}} = \sqrt{0.001659} \approx 0.04073 $$

Rounding to three decimal places gives \(0.041\).

Answer:

Question 1

• (A) The standard deviation of the sampling distribution is \(\sigma\). (Correct answer)
• (B) The sampling distribution is generated by repeatedly taking samples of size \(n\) and computing the sample means.
• (C) The mean of the sampling distribution is \(\mu\).
• (D) The sampling distribution is approximately normal whenever the sample size is sufficiently large (\(n \ge 30\)).

Question 2

• (A) .21; .02
• (B) .21; .407
• (C) .21; .041 (Correct answer)
• (D) .21; .008