Activity 16

MATH 216: Statistical Thinking

Statistical Estimation and Unbiasedness

Time Allocation: 15 minutes total (5 min reading, 10 min individual work)

Part 1: Conceptual Understanding (3 minutes)

Instructions: Answer the following questions about statistical estimation:

  1. Why is unbiasedness important in statistical estimation?
  1. What is the trade-off between bias and variance in estimators?
  1. What does standard error measure in sampling distributions?

Sampling distributions of sample mean and sample median

Part 2: Estimator Comparison (4 minutes)

Scenario: Steel sheets have thickness following uniform distribution (150-200 mm). Two statistics estimate the population mean:

  • Sample mean (\(\bar{x}\))
  • Sample median (M)

Based on sampling distributions, which estimator is better and why?

Calculate expected properties:

  • True population mean = mm
  • Expected bias of sample mean =
  • Expected bias of sample median =
  • Which estimator is unbiased?
  • Which estimator has lower variance?
Comparison of Estimator Properties
Estimator Mean Std Error Bias MSE
Sample Mean 175.13 4.54 0.13 20.60
Sample Median 174.70 7.03 -0.30 49.41

Bias-variance tradeoff visualization

Part 3: Real-World Application (3 minutes)

Quality Control Scenario: A factory needs to estimate average steel thickness for quality assurance.

Decision Analysis:

  • Would you prefer an unbiased estimator with high variance or a slightly biased estimator with low variance? Why?
  • What practical considerations might influence this choice in a manufacturing setting?

Critical Thinking: When might the median be preferred over the mean, even if it’s biased?

Demonstrating median robustness to outliers