Advanced Statistics Test Questions on Estimator Unbiasedness

📚 Quick Study Guide

• 🎯 An estimator is unbiased if its expected value equals the true population parameter. Mathematically, $E(\hat{\theta}) = \theta$, where $\hat{\theta}$ is the estimator and $\theta$ is the true parameter.
• 🧪 Bias is the difference between the expected value of the estimator and the true parameter: $Bias(\hat{\theta}) = E(\hat{\theta}) - \theta$. For an unbiased estimator, $Bias(\hat{\theta}) = 0$.
• 🔢 The sample mean, $\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i$, is an unbiased estimator of the population mean, $\mu$, if the sample is randomly selected.
• 📊 The sample variance, $S^2 = \frac{1}{n-1}\sum_{i=1}^{n} (X_i - \bar{X})^2$, is an unbiased estimator of the population variance, $\sigma^2$. However, the biased version using $n$ in the denominator is biased.
• 💡 To check for unbiasedness, calculate the expected value of the estimator and compare it to the true parameter. If they are equal, the estimator is unbiased.

Practice Quiz

1. Question 1: Which of the following statements best describes an unbiased estimator?
   1. An estimator that always equals the true parameter.
   2. An estimator whose expected value equals the true parameter.
   3. An estimator with a small variance.
   4. An estimator that is always positive.
2. Question 2: What is the bias of an unbiased estimator $\hat{\theta}$ for a parameter $\theta$?
   1. Always positive.
   2. Equal to the variance of $\hat{\theta}$.
   3. Zero.
   4. Equal to the standard deviation of $\hat{\theta}$.
3. Question 3: If $E(\hat{\theta}) = \theta + c$, where $c$ is a non-zero constant, what is the bias of $\hat{\theta}$?
   1. $\theta$.
   2. $c$.
   3. $\theta + c$.
   4. $0$.
4. Question 4: Which of the following is generally an unbiased estimator of the population mean?
   1. The sample median.
   2. The sample mode.
   3. The sample mean.
   4. The trimmed sample mean.
5. Question 5: Why is the sample variance typically calculated with $(n-1)$ in the denominator instead of $n$?
   1. To increase the variance.
   2. To make it an unbiased estimator of the population variance.
   3. To decrease the standard deviation.
   4. To simplify calculations.
6. Question 6: Suppose you have an estimator $\hat{\theta}$ such that $E(\hat{\theta}) = 2\theta$. Is this estimator unbiased?
   1. Yes, always.
   2. Yes, if $\theta = 0$.
   3. No.
   4. Yes, if $\theta = 1$.
7. Question 7: If an estimator has a bias of zero, what does this imply about its consistency?
   1. It is always consistent.
   2. It is always inconsistent.
   3. It provides information only about its unbiasedness, not consistency.
   4. It is consistent only if the variance goes to infinity.