Question

Part A: Calculate the standard deviation of the sampling distribution of X, given a population mean of 27.1, a standard deviation of 8.4, and a sample size of 10. Show your work. (5 points)

Part B: Determine the required sample size for the standard deviation of the sampling distribution to be 2.8, assuming a population mean of 27.1 and a standard deviation of 8.4. Show your work. (5 points)

Answer 1

To calculate the standard deviation of the sampling distribution of X, divide the population standard deviation by the square root of the sample size. To determine the required sample size for a desired standard deviation, divide the population standard deviation by the desired standard deviation and square the result.

Step-by-step explanation:

Part A:

To calculate the standard deviation of the sampling distribution of X, we can use the formula:

Standard Deviation of Sampling Distribution (σx) = Population Standard Deviation (σ) / √Sample Size (n)

Given that the population mean (μ) is 27.1, the population standard deviation (σ) is 8.4, and the sample size (n) is 10, we can plug in these values into the formula:

σx = 8.4 / √10

Simplifying, we get:

σx ≈ 8.4 / 3.162

σx ≈ 2.660

Therefore, the standard deviation of the sampling distribution of X is approximately 2.660.

Part B:

To determine the required sample size for the standard deviation of the sampling distribution to be 2.8, we can rearrange the formula:

Sample Size (n) = (Population Standard Deviation (σ) / Desired Standard Deviation (σx))^2

Given that the population mean (μ) is 27.1, the population standard deviation (σ) is 8.4, and the desired standard deviation (σx) is 2.8, we can plug in these values into the formula:

n = (8.4 / 2.8)^2

Simplifying, we get:

n = 9^2

n = 81

Therefore, the required sample size for the standard deviation of the sampling distribution to be 2.8 is 81.