All Free Statistics MCQs with Answers
Every Statistics question in the bank, across all chapters, each with the correct answer and a written explanation. Free and unlimited, with no account needed.
545 questions · page 28 of 55
271. If E(θˆ)=θ, then θˆ is said to be_______________?
- A. Unbiased
- B. Consistent
- C. Sufficient
- D. Efficient
Explanation: An estimator is unbiased when its expected value equals the parameter it estimates, so E(θ̂)=θ is the defining condition. Consistency concerns convergence as the sample size increases, not equality of expectations.
Correct answer: Unbiased272. If the conditional distribution of X1,X2,.....Xn given S=s, does not depends on θ, for any value of S=s the statistics S=s(X1,X2,.....Xn) is called______________?
- A. Unbiased
- B. Sufficient
- C. Consistent
- D. Efficient
Explanation: By the conditional-distribution characterization of sufficiency, S is sufficient for θ when the distribution of the sample given S=s does not depend on θ. The other properties concern bias, convergence, or variance efficiency.
Correct answer: Sufficient273. A set of jointly sufficient statistic is defined to be minimal sufficient if and only if_______________?
- A. It is a function of some other set of sufficient statistics.
- B. It is a function of every other set of sufficient
- C. It is a function of any sufficient statistics in the set.
- D. It is not a function of every other set of sufficient statistics.
Explanation: A minimal sufficient statistic contains the least information needed for sufficiency and can be expressed as a function of every other sufficient statistic. Option a reverses this relationship and would not characterize minimal sufficiency.
Correct answer: It is a function of every other set of sufficient274. For a biased estimator θˆ of θ, which one is correct_______________?
- A. MSE(θˆ)=SD(θˆ)+Bias
- B. MSE(θˆ)=Var(θˆ)+Bias2
- C. MSE(θˆ)=SD(θˆ)+Bias2
- D. MSE(θˆ)=Var(θˆ)+Bias
Explanation: The mean squared error decomposes into variance plus squared bias: MSE(θ̂)=Var(θ̂)+[Bias(θ̂)]². Bias itself may be positive or negative, so it cannot be added without squaring.
Correct answer: MSE(θˆ)=Var(θˆ)+Bias2275. Consistency of an estimator can be checked by comparing____________?
- A. Standard Deviation
- B. Mean
- C. mean Square Error
- D. Variance
Explanation: Consistency is commonly examined through the estimator's variance as the sample size increases: a consistent estimator should have variance tending to zero, together with vanishing bias. Mean square error is also useful for overall estimator quality, but variance is the expected criterion in this question.
Correct answer: Variance276. Criteria to check a point estimator to be good are_____________?
- A. Consistency
- B. All Above
- C. Unbiasedness
- D. Efficiency
Explanation: A good point estimator is judged by several properties, especially unbiasedness, consistency, and efficiency. Since all three are listed, “All above” is the appropriate choice.
Correct answer: All Above277. 95% confidence limits for P are _________________?
- A. P±2.18√p2q2/n
- B. P±1.65(pq/n)
- C. P±1.96 p2q2/√n
- D. P±1.96√p2q2/n
Explanation: For a population proportion, the 95% confidence interval is estimated proportion ± 1.96√(pq/n), where q = 1 − p. Option d is evidently intended, although the powers or subscripts in the displayed formula appear garbled.
Correct answer: P±1.96√p2q2/n278. The Estimation and testing of hypothesis are main branches of ________________?
- A. Sampling
- B. Probability distribution
- C. Regression
- D. Statistical Inference
Explanation: Statistical inference uses sample information to draw conclusions about a population. Its two main branches are estimation and hypothesis testing.
Correct answer: Statistical Inference279. ___________ is a value of sample statistics?
- A. Estimate
- B. Estimator
- C. Probability
- D. Numerical
Explanation: An estimate is the numerical value obtained from a sample statistic for estimating a population parameter. The estimator is the rule or statistic itself, so it is not the particular value.
Correct answer: Estimate280. Since E(X) _______X is said to be unbiased estimator of the population mean?
- A. Q
- B. X
- C. Ux
- D. u
Explanation: A sample mean is unbiased for the population mean when its expected value equals that population mean, written E(X̄) = μX. The symbol rendered as “u” in option d is evidently the Greek letter μ.
Correct answer: u