5.05a-b Sampling distributions and unbiased estimates Quiz: OCR Further Maths, Unit 2

The 20 questions

1. What is the unbiased estimator for a population mean?

   • Sample mean
   • Population variance
   • Sample range
   • Sample variance

2. Which denominator is used to calculate an unbiased estimator of population variance?

   • n + 1
   • n
   • sqrt(n)
   • n - 1

3. If the expected value of an estimator equals the population parameter, what is it called?

   • Efficient
   • Unbiased
   • Consistent
   • Biased

4. What is the variance of the sample mean for sample size n and variance sigma^2?

   • sigma / n
   • sigma^2 / sqrt(n)
   • sigma^2 / n
   • n sigma^2

5. What is the standard error of the sample mean for sample size n?

   • sqrt(n) / sigma
   • sigma^2 / n
   • sigma / sqrt(n)
   • sigma / n

6. A sample of 25 items gives sample variance 12. What is the unbiased population variance estimate?

   • 12.5
   • 12.0
   • 13.0
   • 11.52

7. Which theorem states the sample mean becomes normally distributed for large sample sizes?

   • Central Limit Theorem
   • Markov's theorem
   • Chebyshev's inequality
   • Bayes' theorem

8. When the expectation of an estimator equals the parameter, what type of estimator is it?

   • Population mean
   • Biased estimator
   • Unbiased estimator
   • Standard error

9. What is the expectation of the sample mean X-bar from a population with mean mu?

   • n mu
   • mu
   • mu / sqrt(n)
   • mu / n

10. For a population with variance 100 and sample size 25, what is the standard error?

    • 20
    • 10
    • 4
    • 2

11. What happens to the standard error of the sample mean as sample size increases?

    • Equals zero
    • Decreases
    • Stays constant
    • Increases

12. How is the bias of an estimator T for parameter theta calculated?

    • E(T) - theta
    • E(T) + theta
    • E(T^2) - theta^2
    • theta - Var(T)

13. For a sample of size n, what is the expected value of sum of squared deviations?

    • sigma^2 / n
    • n sigma^2
    • (n - 1)sigma^2
    • (n + 1)sigma^2

14. A sample of 10 values has a sum of 50. What is the unbiased mean estimate?

    • 5.55
    • 5
    • 4.5
    • 50

15. What is the numerical value of bias for an unbiased estimator?

    • n
    • 1
    • 0
    • mu

16. If X is normally distributed with mean mu, what is the distribution of X-bar?

    • N(mu, n sigma^2)
    • N(mu / n, sigma^2)
    • N(mu, sigma^2 / n)
    • N(mu, sigma^2)

17. By what factor must sample size increase to halve the standard error of the mean?

    • 2
    • 4
    • 8
    • 16

18. Why is the sample variance using denominator n a biased estimator of population variance?

    • Overestimates variance
    • Underestimates variance
    • Always equals variance
    • Has infinite variance

19. For the sample values 2, 4, 6, 8, 10, what is the unbiased mean estimate?

    • 5
    • 30
    • 7
    • 6

20. What term describes the probability distribution of a statistic calculated from samples?

    • Normal distribution
    • Sampling distribution
    • Sample frame
    • Population distribution

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