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IB Mathematics: Applications and Interpretation HL · 13.4: Hypothesis Testing · 13.4.16

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Testing a Poisson mean Problem set B

Scale the rate to the period observed, write hypotheses, find p-values or critical regions with the 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. For each test, X ~ Po(λ). Find the p-value and state the conclusion.

    1. (a) H0: λ = 15, H1: λ < 15, significance level 5%, observed value x = 10.
    2. (b) H0: λ = 6.4, H1: λ < 6.4, significance level 5%, observed value x = 2.

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

    1. (a) H0: λ = 3.5, H1: λ > 3.5, significance level 5%, observed value x = 8.
    2. (b) H0: λ = 14, H1: λ > 14, significance level 10%, observed value x = 20.

  3. Find the critical region for each test.

    1. (a) X ~ Po(λ), H0: λ = 18, H1: λ < 18, significance level 5%.
    2. (b) X ~ Po(λ), H0: λ = 6.5, H1: λ < 6.5, significance level 10%.

  4. Find the critical region for each test.

    1. (a) X ~ Po(λ), H0: λ = 15, H1: λ > 15, significance level 5%.
    2. (b) X ~ Po(λ), H0: λ = 0.8, H1: λ > 0.8, significance level 2%.

In context

  1. A machine has averaged 9 faults per week. After a service, the engineer thinks the average rate has decreased. Over the next 4 weeks there are 25 faults. Test his belief at the 5% significance level.

    1. (a) Write down hypotheses, defining λ and X.
    2. (b) Find the p-value.
    3. (c) State the conclusion in context.

  2. A toll gate has averaged 2.5 motorbikes per hour. After a new road opens, the operator thinks the average rate has increased. In a 4-hour period 15 motorbikes pass. Test his belief at the 10% significance level.

    1. (a) Write down hypotheses, defining λ and X.
    2. (b) Find the p-value.
    3. (c) State the conclusion in context.

  3. A bakery sells an average of 16 sourdough loaves per hour in the morning. After a rival bakery opens next door, the owner thinks the average rate has decreased. He counts the loaves sold in one morning hour. Use a 5% significance level.

    1. (a) Define X and write down hypotheses.
    2. (b) Find the critical region.
    3. (c) There are 10 loaves. State the conclusion.

  4. Find the critical region for each binomial test.

    1. (a) X ~ B(90, p), H0: p = 0.12, H1: p > 0.12, significance level 5%.
    2. (b) X ~ B(50, p), H0: p = 0.3, H1: p > 0.3, significance level 10%.

Stretch

  1. A quarry has averaged 0.9 rockfalls per week. A geologist thinks the average rate has increased after heavy rain. He counts the rockfalls over a 10-week period and tests at the 5% significance level.

    1. (a) State two conditions that must hold for the Poisson distribution to be a suitable model.
    2. (b) Write down hypotheses and find the critical region for the number of rockfalls in 10 weeks.
    3. (c) There are 16 rockfalls in the 10 weeks. State the conclusion in context.

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