1. Which of the following best describes a sampling distribution?

A. The distribution of population values B. The distribution of the values of a statistic across all samples from a population C. The distribution of sample data points D. The distribution of errors from an estimator

2. The critical value used in a 95% confidence interval typically is:

A. 1.28 B. 1.645 C. 1.96 D. 2.58

3. An estimator is said to be consistent if:

A. Its variance increases with sample size B. Its expected value equals the population parameter for all sample sizes C. Its value converges to the true population parameter as sample size increases D. It uses all data without leaving out any observations

4. When constructing a confidence interval for the difference between two population means with unknown variances and small sample sizes (n1, n2 < 30), which distribution is used for the critical value?

A. Normal distribution B. Chi-square distribution C. Student’s t distribution D. F-distribution

5. What is the formula for the standard deviation of the difference between two sample proportions \( \hat{P}_1 \) and \( \hat{P}_2 \)?

A. \( \sqrt{\frac{\hat{P}_1 (1 - \hat{P}_1)}{n_1} - \frac{\hat{P}_2 (1 - \hat{P}_2)}{n_2}} \) B. \( \sqrt{\frac{\hat{P}_1 (1 - \hat{P}_1)}{n_1} + \frac{\hat{P}_2 (1 - \hat{P}_2)}{n_2}} \) C. \( \frac{\hat{P}_1 - \hat{P}_2}{\sqrt{n_1 + n_2}} \) D. \( \hat{P}_1 - \hat{P}_2 \)

6. What does a point estimator provide in estimation theory?

A. A range of possible values for a parameter B. A single best estimate of a population parameter from sample data C. The exact value of the population parameter D. The maximum value observed in a sample

7. Which of the following best defines the consistency property of a point estimator?

A. The estimator always underestimates the parameter B. The estimator gets closer to the parameter as sample size increases C. The estimator has the smallest variance among all unbiased estimators D. The estimator is normally distributed

8. What does sufficiency mean in the context of a point estimator?

A. It is unbiased B. It has the lowest variance C. It contains all information from the data relevant to estimating the parameter D. It is consistent as sample size increases

9. Which distribution is used when estimating a population mean with unknown variance and a small sample size?

A. Normal distribution B. Chi-square distribution C. t-distribution D. F-distribution

10. Which property of a point estimator ensures its expected value equals the true population parameter?

A. Consistency B. Efficiency C. Sufficiency D. Unbiasedness

11. What does an efficient estimator mean?

A. It has the largest variance B. It contains all possible data of the population C. It has the smallest variance among unbiased estimators D. It is biased but consistent

12. Which of the following is NOT true about the t-distribution?

A. It is symmetric about zero B. It approaches the normal distribution as degrees of freedom increase C. It has heavier tails than the normal distribution D. It requires known population variance

13. A 95% confidence interval implies that:

A. 95% of the sample values lie within this interval B. 95% of all possible samples will have intervals containing the true parameter C. There is a 95% probability that the sample mean is within the interval D. The population mean is definitely inside the interval

14. Which of the following is NOT a use of estimation theory?

A. Testing if a sample could come from a population B. Estimating population parameters via interval estimates C. Calculating probabilities of outcomes in a sample D. Working out confidence intervals

15. How is the 100(1−𝛼)% confidence interval for the difference in two population proportions \( P_1 - P_2 \) expressed?

A. \( (\hat{P}_1 + \hat{P}_2) \pm Z_{\frac{\alpha}{2}} \sigma_{\hat{P}_1 - \hat{P}_2} \) B. \( (\hat{P}_1 - \hat{P}_2) \pm Z_{\frac{\alpha}{2}} \sigma_{\hat{P}_1 - \hat{P}_2} \) C. \( (\hat{P}_1 - \hat{P}_2) \pm t_{\frac{\alpha}{2}} \sigma_{\hat{P}_1 + \hat{P}_2} \) D. \( (\hat{P}_1 - \hat{P}_2) \pm Z_{\alpha} \sqrt{\hat{P}_1 (1-\hat{P}_1) + \hat{P}_2 (1-\hat{P}_2)} \)

16. When is a point estimator said to be unbiased?

A. When its value gets closer to the parameter as sample size increases B. When the variance of the estimator is zero C. When its expected value equals the population parameter being estimated D. When it contains all the information about the parameter

17. What statistical value is used to determine the critical value in a confidence interval when the population variance is unknown and the sample size is small?

A. Z value B. Chi-square value C. t value D. F value

18. How does the shape of the t-distribution change as sample size increases?

A. It becomes more skewed to the right B. It becomes identical to the normal distribution C. It becomes more discrete D. It flattens and loses its symmetry

19. When constructing a confidence interval for a proportion, which component is necessary?

A. The sample variance B. The sample proportion and sample size C. The population mean D. The t-score for small samples

20. In constructing confidence intervals for the mean, why does the t-distribution often replace the normal distribution when sample sizes are small?

A. Because population is not normal B. Because the sample variance is used as an estimate for population variance C. Because the sample mean is unknown D. Because the sample size exceeds 30

21. When comparing two unbiased estimators, the one with lesser variance is said to be:

A. Consistent B. Efficient C. Sufficient D. Biased

22. When constructing a confidence interval for the difference between two population means with small samples and unknown variances which are assumed equal, the critical value is from:

A. The standard normal table (Z) B. The chi-square distribution C. The t-distribution with pooled degrees of freedom D. The F-distribution

23. What does the term “confidence level” represent in interval estimation?

A. Probability the true parameter lies outside the interval B. Percentage of intervals that contain the parameter if sampling is repeated infinitely C. The percentage of the sample data lying inside the interval D. Chance that sample mean equals population mean

24. The main shortcoming of point estimates is that:

A. They require large samples B. They do not provide information about how close the estimate is to the true value C. They are always biased D. They are too complex to compute

25. In the example comparing Business administration and Accounting students' pass rates, what does the positive confidence interval for difference in proportions suggest?

A. The proportions are equal B. Business admin students have a higher passing proportion than Accounting students C. Accounting students outperform Business admin D. No conclusion can be made

26. Which of the following is true about the standard error?

A. It increases as sample size increases B. It measures the variability of the sample mean from the population mean C. It is the variance of the sample data D. It states the exact difference between sample and population parameters

27. When constructing a confidence interval for the difference between two proportions, the combined standard error depends on:

A. The variance of one sample only B. Variances and sample sizes of both samples C. The total number of successes only D. Population parameters

28. What is the primary use of sampling distribution in estimation theory?

A. To determine the distribution of individual data points B. To understand the distribution of a statistic obtained from samples C. To estimate the population mean directly D. To calculate population variance

29. The pooled standard deviation is used when:

A. Population variances are known B. Variances are unknown but assumed equal and both sample sizes are small C. Population variances are unequal D. Estimating a single population proportion

30. What does the \( t_{\frac{\alpha}{2}}(n-1) \) term represent in the confidence interval formula for a small sample?

A. Z-distribution critical value B. Chi-square distribution value C. Student's t distribution value with (n-1) degrees of freedom D. Sample mean

31. Which formula represents the confidence interval for a population mean when the population standard deviation is known and the sample size is large (n ≥ 30)?

A. \( \bar{X} \pm Z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} \) B. \( \bar{X} \pm t_{\frac{\alpha}{2}} \frac{S}{\sqrt{n}} \) C. \( \bar{X} \pm Z_{\alpha} \frac{\sigma}{n} \) D. \( \bar{X} \pm t_{\alpha} \frac{\sigma}{\sqrt{n}} \)

32. The formula for the confidence interval for a population proportion is:

A. Sample proportion ± Z * sqrt(sample proportion * (1 - sample proportion) / n) B. Sample mean ± t * (S / sqrt(n)) C. Population proportion ± standard deviation D. Sample mean ± Z * standard deviation

33. Given a sample proportion \( \hat{P} = \frac{x}{n} \), the confidence interval for the population proportion \( P \) is:

A. \( \hat{P} \pm Z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{P}(1-\hat{P})}{n}} \) B. \( \hat{P} \pm t_{\frac{\alpha}{2}} \sqrt{\frac{\hat{P}}{n}} \) C. \( \hat{P} \pm Z_{\frac{\alpha}{2}} \frac{\sigma}{n} \) D. \( \hat{P} \pm t_{\frac{\alpha}{2}} \sigma \)

34. When the population standard deviation is unknown and sample size is small (n < 30), the confidence interval for the population mean is:

A. \( \bar{X} \pm Z_{\frac{\alpha}{2}} \frac{S}{\sqrt{n}} \) B. \( \bar{X} \pm t_{\frac{\alpha}{2}} (n-1) \frac{S}{\sqrt{n}} \) C. \( \bar{X} \pm t_{\frac{\alpha}{2}} (n-1) \frac{\sigma}{n} \) D. \( \bar{X} \pm t_{\frac{\alpha}{2}} \frac{S}{\sqrt{n}} \)

35. What is the difference between point estimate and point estimator?

A. Point estimator is a sample statistic; point estimate is its computed value from data B. Point estimate is a range of values; point estimator is the mean of the range C. Both are exactly the same D. Point estimator always refers to population parameter

36. What is the purpose of interval estimation?

A. To offer a single value estimate for a parameter B. To construct a range within which the population parameter is expected to lie C. To estimate the sample mean accurately D. To measure the standard deviation of a population

37. In interval estimation, the margin of error is influenced by:

A. The sample mean only B. The number of successes in a sample C. The standard deviation and sample size D. The population parameter itself

38. The degrees of freedom \( v \) used in the two-sample t-confidence interval for difference of means with pooled variance is:

A. \( n_1 + n_2 \) B. \( n_1 + n_2 - 2 \) C. \( \min(n_1, n_2) - 1 \) D. \( \max(n_1, n_2) - 1 \)

39. What happens to the margin of error if the sample size increases, assuming other variables remain constant?

A. Margin of error increases B. Margin of error decreases C. Margin of error stays the same D. Margin of error doubles

40. What is the role of the standard error in confidence interval estimation?

A. It measures spread of population data B. It measures variability of the point estimate from the true parameter C. It is the difference between upper and lower confidence limits D. It determines the sample size

41. When the population variance is known and sample sizes are large (n≥30), the confidence interval for the mean uses which distribution?

A. Chi-square B. Normal (Z) distribution C. t-distribution D. Poisson distribution

42. When comparing two population means with unknown and unequal variances, which method is preferred?

A. Use pooled variance method B. Use Z-distribution C. Use Welch’s t-test (not covered in detail here) D. Use chi-square test for means

43. What is sufficiency in the context of an estimator?

A. The estimator is always less than the parameter B. The estimator contains all information about the parameter from the sample C. The estimator achieves minimum variance D. The estimator is unbiased

44. What is the definition of a point estimator?

A. A range of values in which a population parameter lies B. A single statistic obtained from sample observations to estimate a population parameter C. The probability distribution of a statistic D. The variance of a population

45. What is the standard error of the sample mean defined as?

A. \( \sigma \times n \) B. \( \sigma / \sqrt{n} \) C. \( \sigma \times \sqrt{n} \) D. \( S / n \)

46. If you increase the confidence level from 95% to 99%, what happens to the width of the confidence interval?

A. It becomes narrower B. It becomes wider C. It remains the same D. It depends on the sample size, not the confidence level

47. The degree of freedom when pooling variances from two samples of sizes n1 and n2 is:

A. n1+n2 B. n1+n2-1 C. n1+n2-2 D. (n1+n2)/2

48. In the formula for the confidence interval of the difference between two means with unknown variances and small sample sizes, the margin of error is:

A. \( t_{\frac{\alpha}{2}, v} \times \sigma \) B. \( Z_{\frac{\alpha}{2}} \times S_p \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} \) C. \( t_{\frac{\alpha}{2}, v} \times S_p \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} \) D. \( Z_{\frac{\alpha}{2}} \times \sqrt{S_1^2 + S_2^2} \)

49. The pooled standard deviation \( S_p \) when estimating difference of means with unknown but assumed equal variances is computed by:

A. \( \sqrt{\frac{S_1^2 + S_2^2}{2}} \) B. \( \sqrt{\frac{(n_1 - 1) S_1^2 + (n_2 -1) S_2^2}{n_1 + n_2 -2}} \) C. \( \sqrt{S_1^2 + S_2^2} \) D. \( \frac{S_1 + S_2}{2} \)

50. For a population proportion, what condition justifies using the normal approximation to calculate confidence intervals?

A. Sample size is small B. Both np and n(1-p) are sufficiently large (usually greater than 5 or 10) C. The population is normally distributed D. The variance is known