Biased and Unbiased Point Estimates - AP Statistics

1. Is an estimator with higher variance less reliable?

   Yes, higher variance implies less reliability. Higher variance means more variability around target.

2. State the formula for sample proportion.

   $\frac{x}{n}$, where $x$ is the number of successes. Counts successes divided by total trials.

3. Which condition ensures an unbiased estimate?

   Expected value equals the true parameter. No systematic over or under estimation occurs.

4. Find and correct the bias in estimating $\theta$ with $\theta + 1$.

   Correct by using estimator $\theta$. Subtract 1 to remove systematic overestimation.

5. Why is unbiasedness important in estimation?

   Ensures estimates are centered around true parameter. Prevents systematic over or under estimation.

6. What is the bias of using $n$ in sample variance calculation?

   Underestimates true variance, biased. Creates systematic underestimation of true variance.

7. What estimator property improves with increasing sample size?

   Consistency. Estimator precision improves with larger samples.

8. What is the bias of the sample mean for population mean?

   Zero, sample mean is unbiased. Sample mean has no systematic error.

9. Identify the unbiased estimator for population proportion.

   Sample proportion $\frac{x}{n}$. Expected value equals true population proportion $p$.

10. What is a point estimate?

    A single value estimate for a population parameter. Provides best guess of unknown population parameter.

11. What is the relationship between bias and mean squared error?

    MSE = Variance + Bias$^2$. Decomposition shows bias contributes to total error.

12. Can a biased estimator converge to the true parameter?

    Yes, if it is consistent. Asymptotic convergence overcomes initial bias.

13. Define unbiased point estimate.

    An estimate whose expected value equals the true parameter. No systematic error in long run.

14. State the formula for sample mean.

    $\bar{x} = \frac{\textstyle \text{Sum of all sample values}}{n}$. Sum divided by sample size gives average.

15. Is the sample mode an unbiased estimator for population mean?

    No, mode does not equate to mean. Mode represents most frequent value, not average.

16. Is an estimator unbiased if its bias is zero?

    Yes, zero bias means unbiased. Zero bias is definition of unbiased estimator.

17. Which estimate is unbiased for population variance?

    Sample variance $s^2 = \frac{\textstyle \text{SS}}{n-1}$. Dividing by $n-1$ corrects for degrees of freedom.

18. Why is $s^2 = \frac{\text{SS}}{n-1}$ unbiased?

    Accounts for degrees of freedom, matches true variance. Corrects for lost degree of freedom from estimating mean.

19. Find the bias in the estimate: sample variance $s^2 = \frac{\text{SS}}{n}$.

    Bias: underestimates population variance. Dividing by $n$ instead of $n-1$ creates downward bias.

20. What measure do statisticians use to quantify bias?

    Expected value of the estimator minus true parameter. Measures systematic deviation from true value.

21. Name a situation where an estimator can be unbiased but inefficient.

    Estimator has high variance. Large variability reduces precision despite unbiasedness.

22. Is the sample mean consistent for estimating population mean?

    Yes, the sample mean is consistent. Converges to $\mu$ by Law of Large Numbers.

23. What does it mean if an estimator is 'consistent'?

    Estimator converges to true parameter as sample size increases. Approaches true value with infinite sample size.

24. What is the bias in using sample median for population mean?

    Median may systematically differ from mean. Creates bias except in symmetric populations.

25. What is the impact of sample size on bias?

    Larger samples may reduce bias, not eliminate it. Bias is property of estimator, not sample size.

26. What is the role of sample size in unbiased estimation?

    Larger samples improve estimate accuracy, not bias. Size affects precision, not bias direction.

27. Are all consistent estimators also unbiased?

    No, consistency does not imply lack of bias. Consistency and bias are independent properties.

28. Why might a point estimate be biased?

    Due to systematic error or sample selection issues. Flawed methodology or non-random sampling causes bias.

29. What is a common source of bias in surveys?

    Non-response bias. Missing responses skew results systematically.

30. Is the median an unbiased estimator for mean?

    No, median may not equal mean. Median only equals mean in symmetric distributions.

31. What characterizes a biased point estimate?

    An estimate systematically different from the parameter. Contains systematic error, not centered on true value.

32. Determine if sample range is an unbiased estimator for range.

    No, sample range underestimates population range. Sample cannot capture full population range.

33. Identify an unbiased estimator for population parameter.

    Estimator with expected value equal to the parameter. Expected value must match the true parameter.

34. State the formula for sample variance.

    $s^2 = \frac{\text{SS}}{n-1}$. Uses $n-1$ to correct for degrees of freedom.

35. What does it mean if an estimator is 'efficient'?

    It has the smallest variance among unbiased estimators. Minimum variance unbiased estimator is optimal.

36. Does increasing sample size always eliminate bias?

    No, some biases persist regardless of sample size. Structural biases persist regardless of sample size.

37. Identify the unbiased estimate for population mean.

    Sample mean $\bar{x}$. Expected value equals population mean $\mu$.

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