Question

question 1 which of the following statements about the sampling distribution of the sample mean, x - bar, is true? check all that apply. select all that apply. 10 points a. the distribution is normal regardless of the shape of the population distribution, as long as the sample size, n, is large enough. b. the distribution is normal regardless of the sample size, as long as the population distribution is normal. c. the distributions mean is the same as the population mean. d. the distributions standard deviation is smaller than the population standard deviation. question 2 pictured below (in scrambled order) are three histograms. one of them represents a population distribution. the other two are sampling distributions of x - bar: one for sample size n = 5 and one for sample size n = 40. select one answer. 10 points

Explanation:

• A: The Central Limit Theorem states that for a sample of size \(n\) from any population (regardless of its shape), when \(n\) is large (usually \(n\geq30\)), the sampling - distribution of the sample mean \(\bar{x}\) is approximately normal.
• B: If the population is normally distributed, then the sampling distribution of the sample mean \(\bar{x}\) is normal for any sample size \(n\).
• C: The mean of the sampling distribution of the sample mean, denoted as \(\mu_{\bar{x}}\), is equal to the population mean \(\mu\), i.e., \(\mu_{\bar{x}}=\mu\).
• D: The standard deviation of the sampling distribution of the sample mean (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size. Since \(n > 1\), \(\sigma_{\bar{x}}<\sigma\).

Answer:

A. The distribution is normal regardless of the shape of the population distribution, as long as the sample size, n, is large enough.
B. The distribution is normal regardless of the sample size, as long as the population distribution is normal.
C. The distribution's mean is the same as the population mean.
D. The distribution's standard deviation is smaller than the population standard deviation.