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IB Mathematics: Applications and Interpretation HL · 13.4: Hypothesis Testing · 13.4.15

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Testing a proportion Problem set B

Define X, write hypotheses, find p-values or critical regions with the binomial cumulative probability, and conclude in context.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Write down suitable hypotheses for each test. Define p in words.

    1. (a) A cinema says 35% of its tickets are sold online. The owner thinks fewer tickets are sold online now.
    2. (b) A farmer says 12% of her eggs are cracked on delivery. A shop thinks that more eggs are cracked.
    3. (c) A die is said to show a six with probability 1/6. A player thinks it shows a six less often.

  2. For each test, X ~ B(n, p). Find the p-value and state the conclusion.

    1. (a) H0: p = 0.55, H1: p < 0.55, n = 30, significance level 5%, observed value x = 12.
    2. (b) H0: p = 0.1, H1: p > 0.1, n = 40, significance level 5%, observed value x = 8.

  3. Find the critical region for each test.

    1. (a) X ~ B(50, p), H0: p = 0.6, H1: p < 0.6, significance level 5%.
    2. (b) X ~ B(45, p), H0: p = 0.3, H1: p < 0.3, significance level 1%.

  4. Find the critical region for each test.

    1. (a) X ~ B(70, p), H0: p = 0.25, H1: p > 0.25, significance level 5%.
    2. (b) X ~ B(40, p), H0: p = 0.7, H1: p > 0.7, significance level 10%.

In context

  1. A school says that 30% of its pupils walk to school. After a new bus route starts, a teacher thinks the proportion who walk has decreased. In a random sample of 45 pupils, 8 walk. Test her belief at the 5% significance level.

    1. (a) Define X and write down hypotheses.
    2. (b) Find the p-value.
    3. (c) State the conclusion in context.

  2. A bus company says that 80% of its buses on one route arrive on time. Commuters think the proportion is lower. They record 60 buses and count those that arrive on time. Use a 5% significance level.

    1. (a) Define X and write down hypotheses.
    2. (b) Find the critical region.
    3. (c) The sample has 39 buses that arrive on time. State the conclusion.

  3. The graph shows the probabilities for X ~ B(30, 0.3). A test of H0: p = 0.3 against H1: p > 0.3 is carried out on a random sample of 30 at the 5% significance level.

    34567891011121314151617180.050.10.150.2xP(X = x)
    1. (a) Find the critical region.
    2. (b) Find the actual probability of rejecting H0 when H0 is true.
    3. (c) Explain why this is not equal to 5%.

Stretch

  1. A multiple-choice quiz is tested for guessing: H0: p = 0.25, H1: p > 0.25, at the 10% significance level. For each number of questions n, find the critical region and its actual probability when H0 is true.

    1. (a) n = 8
    2. (b) n = 16
    3. (c) n = 24

jmaths.xyz · AI HL · 13.4.15 Testing a proportion: the binomial test