Lesson 4B.5.1-4B.5.2

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

20 questions

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Lesson 4B.5.1-4B.5.2, Concepts of standard error and confidence intervals: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 22: Confidence intervals, written with Revision Ninja.

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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 bias of an estimator theta-hat of a parameter theta?

    • E(theta-hat) minus theta
    • The difference between the sample mean and the population mean
    • The variance of the estimator
    • The difference between the largest and smallest estimate

  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 a 95% confidence interval for a population mean represent?

    • 95% of the data values lie within the interval
    • If repeated samples were taken, about 95% of intervals constructed this way would contain the true mean
    • The sample mean lies in the interval with 95% probability
    • There is a 95% probability that the true mean lies in this particular interval

  5. Which property makes an estimator preferable, all else being equal?

    • A large variance, since it captures more of the variation in the data
    • Unbiasedness together with a small variance
    • Deliberate bias, since this always reduces the variance
    • Being the median of the sample, since it is always valid

  6. How are a 95% confidence interval for mu and a two-tailed 5% test of H0: mu = mu0 related?

    • The test rejects H0 when the p-value is greater than 0.05
    • The test rejects H0 at the 5% level exactly when mu0 lies outside the 95% interval
    • They are unrelated, since one is an estimate and the other is a decision
    • The test rejects H0 whenever mu0 lies inside the interval

  7. What happens to a confidence interval for a mean when the sample size n increases, with the confidence level and sigma fixed?

    • The interval stays the same width
    • The interval gets wider, since more data are included
    • The interval gets narrower by a factor of n
    • The interval gets narrower, since the standard error falls as sqrt(n) grows

  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 would make a 95% confidence interval for a population mean narrower?

    • Using a larger population standard deviation
    • Increasing the sample size n
    • Reducing the sample size from 100 to 25
    • Increasing the confidence level to 99%

  13. Estimator T1 is unbiased with variance 4. Estimator T2 has bias 0.5 and variance 1. Which has the smaller mean squared error?

    • Both have the same mean squared error of 4
    • They cannot be compared because bias invalidates the mean squared error
    • T2, with mean squared error 1.25, compared with 4 for T1
    • T1, with mean squared error 4, because it is unbiased

  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. A sample mean X-bar is used to estimate mu from a random sample of size n = 4. Is the single observation X1 an unbiased estimator of mu?

    • No, its expectation is sigma^2
    • Yes, but its variance is larger than that of the sample mean
    • Yes, and its variance is smaller than that of the sample mean
    • No, it is biased downwards

  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. Which statement about a single computed 95% confidence interval is correct?

    • There is a 95% chance that this interval contains the true mean
    • The interval either contains the true mean or it does not; the 95% refers to the long-run procedure
    • The interval contains 95% of the population values
    • The sample mean lies inside this interval 95% of the time

  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. Why might a biased estimator with much smaller variance be preferred to an unbiased one with large variance?

    • Unbiased estimators always have larger variance than biased ones
    • Consistent estimators always have zero bias, so bias is irrelevant
    • The bias cancels out when the sample size is large
    • Its mean squared error, which combines bias and variance, can be smaller overall

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