Interpret Variation across Repeated Samples

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: In fictional simulation 2, repeated random samples of size 100 have means less spread out than samples of size 10. Which interpretation is warranted? Candidate answers: (A) Larger samples can make the sample mean more stable under this model (B) Larger samples guarantee the population mean exactly (C) Sample size removes nonresponse bias (D) The population changed whenever a sample mean changed.
• Independent check inputs: In fictional simulation 3, repeated random samples of size 100 have means less spread out than samples of size 10. Which interpretation is warranted? Candidate answers: (A) Larger samples can make the sample mean more stable under this model (B) Larger samples guarantee the population mean exactly (C) Sample size removes nonresponse bias (D) The population changed whenever a sample mean changed.

Learn the idea

Evidence quality depends on how participants and conditions are selected. Random sampling supports population generalization, random assignment supports treatment comparisons, and uncertainty summaries do not repair biased recruitment or uncontrolled causal differences. Repeated samples from one model can give different means. Larger independent samples commonly produce less variable means under finite-variance conditions, improving precision without guaranteeing an exact population value.

Here is the method on the supplied teaching case. Repeated-sample variability is sampling variation. With independent sampling from the same finite-variance model, larger samples generally reduce the spread of sample means; they do not guarantee exactness or repair a biased design.

Sample size cannot eliminate a biased selection method or nonresponse problem. The supplied simulation compares variation under one model; it does not claim every large sample is representative or every estimate correct.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: In fictional simulation 9, repeated random samples of size 100 have means less spread out than samples of size 10. Which interpretation is warranted? Candidate answers: (A) Larger samples can make the sample mean more stable under this model (B) Larger samples guarantee the population mean exactly (C) Sample size removes nonresponse bias (D) The population changed whenever a sample mean changed. Show the mathematical reason, not only the selected answer. Expected result: Larger samples can make the sample mean more stable under this model. Repeated-sample variability is sampling variation. With independent sampling from the same finite-variance model, larger samples generally reduce the spread of sample means; they do not guarantee exactness or repair a biased design.

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