Testing a Binomial Proportion · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

How surprising is this many heads

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

  1. Jo says a p-value counts only the exact observed result, not anything more extreme. Is Jo right?

    Circle one:   True   False

  2. A binomial test gives p-value 0.0625 with a 5 percent significance level. What is the decision?

    • Do not reject H0 at the 5 percent level
    • Reject H0 at the 5 percent level
    • Accept H0 as proven true
    • Collect more data before deciding anything

  3. A binomial test gives p-value 0.0625 with a 5 percent significance level. What is the decision?

    • Reject H0 at the 5 percent level
    • Accept H0 as proven true
    • Do not reject H0 at the 5 percent level

  4. A test gives p-value 0.03 at the 5 percent level. What is the decision?

    • Reject H0
    • Do not reject H0
    • Halve the level and retest

  5. A coin tossed 5 times shows 5 heads. Under H0: p = 0.5, what is P(X = 5) as a decimal?

    Answer: ______________

  6. One sided test, n = 5, H0: p = 0.5, 5 percent level. Which counts form the critical region?

    • {4, 5}
    • {3, 4, 5}
    • {5}

  7. A coin is tossed 4 times to test H0: p = 0.5 against p > 0.5. All 4 land heads. For this one sided test the p-value is P(X >= 4). What is it as a decimal?

    Answer: ______________

  8. A coin is tossed 4 times to test H0: p = 0.5 against p > 0.5. All 4 land heads. For this one sided test the p-value is P(X >= 4). What is it as a decimal?

    Answer: ______________

  9. After 9 heads in 12 tosses, a test at the 5 percent level gives p about 0.073. What is the verdict in context?

    • Reject H0: the coin is biased
    • Do not reject H0: no convincing evidence of bias
    • Accept H0: the coin is proven fair

  10. One sided test, n = 5, H0: p = 0.5, 5 percent level. Which counts form the critical region?

    • {5}
    • {4, 5}
    • {3, 4, 5}
    • {0}
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Answer key

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

How surprising is this many heads W1-mt_ZJ8MTLYYKE-s1

  1. False · The p-value gathers the observed count plus everything even less compatible with H0. Counting only the exact outcome understates the evidence against the null.
  2. Do not reject H0 at the 5 percent level · 0.0625 sits above 0.05, so the evidence is not strong enough to reject the null at the 5 percent level.
  3. Do not reject H0 at the 5 percent level · 0.0625 sits above 0.05, so the evidence is too weak to reject.
  4. Reject H0 · 0.03 is below 0.05, so the result is significant at the 5 percent level.
  5. 0.03125 · Five heads in a row under a fair coin has probability 0.5^5 = 0.03125.
  6. {5} · Only a count of 5 keeps the tail at or below 0.05.
  7. 0.0625 · Only HHHH of 16 strings is at least as extreme, so 1/16 = 0.0625.
  8. 0.0625 · Under H0 each of the 16 outcome strings is equally likely and only HHHH is at least as extreme, so 1/16 = 0.0625.
  9. Do not reject H0: no convincing evidence of bias · 0.073 sits above 0.05, so the tosses show no convincing evidence of bias.
  10. {5} · P(X = 5) = 1/32 = 0.03125 fits under 0.05, but P(X >= 4) = 6/32 = 0.1875 does not, so only a count of 5 rejects H0.
Worksheet · LightMySky