Math Problem Statement

Which of the following is a biased estimator? That is, which of the following does not target the population parameter?

Solution

The question asks which of the following is a biased estimator, meaning which does not accurately target the population parameter.

Here's an analysis of the options:

  1. Median - The sample median is typically an unbiased estimator of the population median.
  2. Proportion - The sample proportion is an unbiased estimator of the population proportion.
  3. Variance - The sample variance (calculated as 1n1(xixˉ)2\frac{1}{n-1} \sum (x_i - \bar{x})^2) is an unbiased estimator of the population variance. However, if calculated using 1n(xixˉ)2\frac{1}{n} \sum (x_i - \bar{x})^2, it becomes biased.
  4. Mean - The sample mean is an unbiased estimator of the population mean.

Answer: The correct answer is Variance, when it is calculated using 1n\frac{1}{n}, as this does not accurately target the population variance and is considered a biased estimator.

Would you like more details or have any further questions? Here are some related questions you could explore:

  1. Why does dividing by n1n - 1 rather than nn give an unbiased variance estimate?
  2. What are other common examples of biased and unbiased estimators?
  3. How do you calculate the sample mean and population mean?
  4. What is the significance of biased vs. unbiased estimators in statistical inference?
  5. How does sample size affect the accuracy of different estimators?

Tip: Remember, an unbiased estimator has an expected value equal to the parameter it estimates.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Estimators
Population Parameters

Formulas

Variance formula: 1/n or 1/(n-1) ∑(x_i - x̄)^2

Theorems

Biased vs. Unbiased Estimator Theorem

Suitable Grade Level

Grades 11-12 and Undergraduate