due 09/25/2025 8:00 am edt question 1 which of the following statements about the sampling distribution of…

due 09/25/2025 8:00 am edt question 1 which of the following statements about the sampling distribution of the sample mean, x - bar, is true? check all that apply. 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. select all that apply. 10 points

due 09/25/2025 8:00 am edt question 1 which of the following statements about the sampling distribution of the sample mean, x - bar, is true? check all that apply. 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. select all that apply. 10 points

Answer

Brief Explanations:

  • A: The Central - Limit Theorem states that for a sample of size (n) from any population (regardless of the population's shape), when (n) is large enough (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 (\sqrt{n}> 1) for (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.