Sampling Distributions and the Central Limit Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Crowds behave better than individuals

Mathematics · Data & Statistics · ages 17-18
Name ______________________   Date ____________

  1. A population has standard deviation 12. Samples of size 36 are taken. What is the standard error of the sample mean?

    Answer: ______________

  2. A population has standard deviation 20. Samples of size 25 are taken. What is the standard error of the sample mean?

    Answer: ______________

  3. As the sample size grows, what happens to the spread of the sample means?

    • It gets smaller
    • It gets larger
    • It stays exactly the same
    • It matches the population spread exactly

  4. As the sample size grows, what happens to the spread of the sample means?

    • It gets smaller
    • It gets larger
    • It stays exactly the same

  5. A population has standard deviation 9. Samples of size 81 are taken. What is the standard error of the sample mean?

    Answer: ______________

  6. For large samples, what shape does the central limit theorem promise for the sample means?

    • Uniform
    • Skewed to the right
    • Approximately normal

  7. For large samples, what shape does the central limit theorem promise for the sample means?

    • Approximately normal
    • Uniform
    • Skewed to the right
    • Binomial

  8. A population has standard deviation 20. Samples of size 25 are taken. What is the standard error of the sample mean?

    Answer: ______________

  9. A population has standard deviation 20, and samples of 25 give standard error 4. What standard error do samples of 100 give?

    Answer: ______________

  10. Bigger samples move the expected sample mean closer to the truth.

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Crowds behave better than individuals W1-mt_nugaEeUSii-s1

  1. 2 · The standard error is 12 divided by the square root of 36, and 12/6 = 2.
  2. 4 · Root 25 is 5, and 20/5 = 4.
  3. It gets smaller · Averaging over more values cancels out highs against lows, so sample means cluster tighter as n grows.
  4. It gets smaller · Averaging over more values cancels out highs against lows.
  5. 1 · Root 81 is 9, and 9/9 = 1.
  6. Approximately normal · The CLT pulls the distribution of means toward a normal shape for large n.
  7. Approximately normal · The CLT pulls the distribution of means toward a normal shape for large n, whatever shape the population started with.
  8. 4 · Root 25 is 5, and 20/5 = 4.
  9. 2 · Quadrupling the sample halves the spread: 20 over root 100 = 2.
  10. False · The centre stays at the population mean; only the spread shrinks.
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