Lesson 2: 7.2 Sampling Distributions: Center and Variability

Why is a statistic called an "unbiased estimator" if its sampling distribution is centered at the true parameter?

If we want to cut our "estimate error" in half, do we need to double our sample size or do something more drastic?

Does the size of the population (e.g., the U.S. vs. Connecticut) affect how much our sample results vary?

Unbiased estimator

Biased estimator

Variability of a statistic

Precision

Sample size (n)

HSS-IC.A.1: Understand statistics as a process for making inferences about population parameters.

HSS-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error.

The SAT frequently tests the relationship between sample size and margin of error. Students must know that larger samples lead to more precise estimates (less variability) and that the population size is generally irrelevant to the precision of the sample as long as the population is much larger than the sample.

DOK 2: Compare two sampling distributions and determine which statistic is a better estimator based on its bias and variability.

DOK 3: Explain why increasing the sample size from n = 100 to n = 400 reduces the standard deviation of the sampling distribution by half.

The Problem: A pollster wants to estimate the proportion of Connecticut residents who support a new environmental law. In one scenario, they sample n = 100 people. In another, they sample n= 1,000 people.

Task: If the true proportion is p = 0.60, which sampling distribution will have a center closer to 0.60? Which will have a smaller standard deviation? Justify your answer.

The Population Size Myth: Students almost always think that a sample of 1,000 people from a large population (like China) will be "less accurate" than a sample of 1,000 from a small population (like Rhode Island). You must emphasize that variability depends on sample size (n), not the population size (as long as the 10% condition is met).

Bias vs. Variability: Students often use these terms interchangeably. Use the visual of the target: Bias is about where you are aiming; variability is about how consistent your aim is.

Support: Provide a "Sorting Task" with four target images (High Bias/Low Variability, Low Bias/High Variability, etc.) and have students match them to corresponding sampling distribution dot plots.

Extension: Introduce the Square Root Rule. Challenge students to figure out the mathematical relationship between n and variability. Ask: "If I want to divide the variability by 3, what must I do to the sample size?" (n must be 9 times larger).

Teacher assigns examples from the textbook and other resources.

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