4B.5.1-4B.5.2 Concepts of standard error and confidence intervals Quiz: Pearson Edexcel Further Maths, Unit 22

The 20 questions

1. What is the standard error of the sample mean from a random sample of size n from a population with standard deviation sigma?

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

2. What is the formula for the bias of an estimator theta-hat for a parameter theta?

   • E(theta-hat) - theta
   • theta - E(theta-hat)
   • Var(theta-hat)
   • E(theta-hat^2) - theta

3. Which divisor gives an unbiased estimate of the population variance from a sample of size n?

   • n - 1
   • n + 1
   • 2n
   • n

4. What does the percentage in a confidence interval describe in repeated sampling?

   • Sample mean accuracy
   • Long-run success rate
   • Population variance
   • Exact parameter probability

5. Which combination of properties makes a point estimator most desirable?

   • Biased, small variance
   • Unbiased, small variance
   • Unbiased, large variance
   • Biased, large variance

6. A two-tailed test at the 5% level rejects $H_0: \mu = \mu_0$ when $\mu_0$ is:

   • Greater than zero
   • Outside the interval
   • At the centre
   • Inside the interval

7. What happens to the width of a confidence interval as sample size $n$ increases?

   • It stays constant
   • It increases
   • It doubles
   • It decreases

8. A population has standard deviation 12. What is the standard error of the sample mean when n = 36?

   • 6
   • 4
   • 2
   • 1/3

9. A sample is 2, 4, 6, 8. What is the unbiased sample variance s^2?

   • 10/3, about 3.33
   • 5
   • 10
   • 20/3, about 6.67

10. A sample of n = 36 from a normal population with sigma = 6 has mean 52. What is the 95% confidence interval for the population mean?

    • (50.00, 54.00)
    • (50.04, 53.96)
    • (46.08, 57.92)
    • (51.00, 53.00)

11. Using the same sample (n = 36, sigma = 6, mean 52), what is the 99% confidence interval for the mean, to 2 decimal places?

    • (49.42, 54.58)
    • (47.42, 56.58)
    • (49.00, 55.00)
    • (50.04, 53.96)

12. Which change makes a 95% confidence interval for a population mean narrower?

    • Increasing confidence level
    • Increasing sample size
    • Increasing population variance
    • Decreasing sample size

13. Estimator $T_1$ has variance 4, bias 0. $T_2$ has variance 1, bias 0.5. What is the MSE of $T_2$?

    • 1.00
    • 4.00
    • 1.25
    • 1.50

14. A sample of n = 25 from a normal population with sigma = 5 is used to construct a 95% confidence interval for the mean. What is the half-width of the interval?

    • 0.98
    • 3.92
    • 1.96
    • 1.00

15. Compared to the sample mean $\bar{X}$, a single observation $X_1$ from a sample of size $n$ is:

    • Unbiased, smaller variance
    • Unbiased, larger variance
    • Biased, smaller variance
    • Biased, larger variance

16. An estimator divides the sum of squared deviations by n rather than n - 1. For n = 10 and sigma^2 = 20, what is its bias?

    • 2
    • -20
    • -2
    • -0.2

17. What is the smallest whole sample size n for a 95% confidence interval for a mean with half-width at most 1, when sigma = 5?

    • 97
    • 25
    • 10
    • 96

18. For a calculated 95% confidence interval, what is the probability that it contains the true mean $\mu$?

    • Exactly 0.95
    • Either 0 or 1
    • Exactly 0.50
    • Exactly 0.05

19. A sample has mean 100 and standard error 4. What is the 90% confidence interval for the mean, to 2 decimal places?

    • (93.42, 106.58)
    • (92.00, 108.00)
    • (96.08, 103.92)
    • (99.34, 100.66)

20. Which metric combines both bias and variance to evaluate an estimator?

    • Confidence interval
    • Standard error
    • Significance level
    • Mean squared error

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