Question

You are conducting a random sample from a population with a mean (μ) of 99 and a standard deviation (σ) of 10. For each of the following sample sizes, calculate the mean and standard deviation of the sample mean (x) for the sampling distribution. Please round your answers to three decimal places.

(a) For a sample size of n = 11, the mean (μ) of the sampling distribution is ____, and the standard deviation (σ) of the sampling distribution is ____.

(b) For a sample size of n = 37, the mean (μ) of the sampling distribution is ____, and the standard deviation (σ) of the sampling distribution is ____.

(c) For a sample size of n = 100, the mean (μ) of the sampling distribution is ____, and the standard deviation (σ) of the sampling distribution is ____.

Answer 1

For each sample size, the mean (μx) of the sampling distribution is the same as the population mean, which is 99. The sample sizes n=11, n=37, and n=100, the corresponding standard deviations are approximately 3.015, 1.645, and 1.000 respectively.

Step-by-step explanation:

You are conducting a random sample from a population with a mean (μ) of 99 and a standard deviation (σ) of 10. The Central Limit Theorem tells us that the sampling distribution of the sample mean will be normally distributed with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size (n). Using this information, we can calculate the mean and standard deviation of the sample mean (μx and σx) for different sample sizes.

(a) For a sample size of n = 11, the mean (μx) of the sampling distribution is μx = μ = 99, and the standard deviation (σx) of the sampling distribution is σx = σ / √ n = 10 / √ 11 ≈ 3.015.

(b) For a sample size of n = 37, the mean (μx) of the sampling distribution is μx = μ = 99, and the standard deviation (σx) of the sampling distribution is σx = σ / √ n = 10 / √ 37 ≈ 1.645.

(c) For a sample size of n = 100, the mean (μx) of the sampling distribution is μx = μ = 99, and the standard deviation (σx) of the sampling distribution is σx = σ / √ n = 10 / √ 100 = 1.000.