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

Averages settle, shapes turn normal

Mathematics · Data & Statistics · ages 19-20
Name ______________________   Date ____________

  1. Nora flips a coin 100 times and sees mostly tails. She says the CLT makes heads nearly certain on flip 101. Is Nora right?

    Circle one:   True   False

  2. A population has standard deviation 12. Samples of size 36 are drawn. Find the standard deviation of the sample mean.

    Answer: ______________

  3. A population has standard deviation 12. Samples of size 36 are drawn. Find the spread of the sample mean.

    Answer: ______________

  4. Incomes in a town are strongly skewed. Which sample size gives sample means with the most normal shape?

    • 2
    • 5
    • 100

  5. A population has standard deviation 15. Samples of size 25 are drawn. Find the standard deviation of the sample mean.

    Answer: ______________

  6. How do you standardise a sample mean for normal tables?

    • Subtract sigma, then divide by mu
    • Subtract mu, then divide by sigma over root n
    • Add mu, then multiply by root n

  7. A population has standard deviation 15. Samples of size 25 are drawn. Find the spread of the sample mean.

    Answer: ______________

  8. Which pairing states each result correctly?

    • Law of large numbers: averages settle near the mean. CLT: averages are approximately normal for large n.
    • Law of large numbers: single values settle near the mean. CLT: single values are approximately normal.
    • Law of large numbers: averages turn skewed. CLT: averages equal the mean exactly.
    • Law of large numbers: the mean moves toward the sample. CLT: samples stop varying.

  9. Dee earns a skewed income and says the CLT makes her single income normal. What is wrong?

    • The CLT needs a symmetric population first
    • The CLT only works for coin flips
    • The CLT shapes averages, not single observations

  10. A poll quotes a margin of error for a percentage. Which check supports the normal approximation?

    • Expected successes and failures at least 10 each
    • Sample size exactly 30
    • The population itself is normal
LightMySky · lightmysky.comW1-mt_NknG3L5rQp-s1

Answer key

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

Averages settle, shapes turn normal W1-mt_NknG3L5rQp-s1

  1. False · Nora is wrong. The theorem describes averages over many trials, not the next single flip, which is still 50-50.
  2. 2 · Divide sigma by the square root of n: 12 divided by 6 is 2.
  3. 2 · Divide sigma by root n: 12 divided by 6 is 2.
  4. 100 · Larger samples let the normal approximation take over.
  5. 3 · Sigma over root n is 15 divided by 5, which is 3.
  6. Subtract mu, then divide by sigma over root n · Centring by mu and scaling by the spread gives the standard bell.
  7. 3 · Sigma over root n is 15 divided by 5, which is 3.
  8. Law of large numbers: averages settle near the mean. CLT: averages are approximately normal for large n. · The law of large numbers is about settling on the mean; the CLT is about the normal shape of averages.
  9. The CLT shapes averages, not single observations · One observation keeps the population shape; only means smooth out.
  10. Expected successes and failures at least 10 each · Proportions are means of ones and zeros, and need both counts large.
Worksheet · LightMySky