Questions: Sampling Distributions

Question 1

A population has mean μ = 50 and standard deviation σ = 10. For random samples of size n = 25, what is the standard error of the sample mean?

• A. 10
• B. 2
• C. 0.4
• D. 50

Answer: (B)

Standard error = σ/√n = 10/√25 = 10/5 = 2. The standard error measures how much the sample mean varies from sample to sample. It shrinks as n grows because averaging more observations cancels out individual variability.

Question 2

If you increase the sample size from n = 25 to n = 100, the standard error of the sample mean doubles.

• T. True
• F. False

Answer: False

Increasing sample size DECREASES the standard error. Going from n = 25 to n = 100 changes √n from 5 to 10, so the standard error is halved (σ/10 instead of σ/5). A common misconception is that more data means more variability; in fact, averaging more observations reduces the variability of the estimate.

Question 3

What is the difference between the population distribution and the sampling distribution of the sample mean?

Model answer: The population distribution describes how individual observations vary. The sampling distribution of the sample mean describes how the average of a sample (of size n) varies across all possible samples. The sampling distribution has mean μ and standard deviation σ/√n, and is approximately normal for large n regardless of the population's shape.

These are distributions at different levels: one describes raw observations, the other describes a statistic computed from a batch of observations. The distinction is fundamental to inference — when we use x̄ to estimate μ, we are reasoning about the sampling distribution, not the population distribution directly.