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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 A

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

Difficulty: first skills applying them stretch

Skills

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

    1. (a) A bakery says 25% of its customers buy a cake. The manager thinks more customers buy a cake since a new display was added.
    2. (b) A phone maker says 4% of its phones are returned. A reviewer thinks the proportion of returns is lower than this.
    3. (c) A spinner is said to land on red with probability 1/5. A player thinks it lands on red more often.

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

    1. (a) H0: p = 0.4, H1: p < 0.4, n = 25, significance level 5%, observed value x = 5.
    2. (b) H0: p = 0.15, H1: p > 0.15, n = 50, significance level 10%, observed value x = 12.

  3. Find the critical region for each test.

    1. (a) X ~ B(40, p), H0: p = 0.35, H1: p < 0.35, significance level 5%.
    2. (b) X ~ B(60, p), H0: p = 0.5, H1: p < 0.5, significance level 1%.

  4. 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(80, p), H0: p = 0.2, H1: p > 0.2, significance level 10%.

In context

  1. A shop owner says that 20% of customers pay by card. A bank claims that the proportion has increased since it fitted a faster card reader. In a random sample of 60 customers, 17 pay by card. Test the bank's claim 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 nursery says that 45% of its tulip bulbs produce a flower in the first year. After a cold winter, a gardener thinks the proportion is lower. She plants 50 bulbs and counts those that flower. Use a 5% significance level.

    1. (a) Define X and write down hypotheses.
    2. (b) Find the critical region.
    3. (c) The sample has 17 plants with a flower. State the conclusion.

  3. The graph shows the probabilities for X ~ B(25, 0.4). A test of H0: p = 0.4 against H1: p > 0.4 is carried out on a random sample of 25 at the 5% significance level.

    456789101112131415161718190.050.10.150.2xP(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 coin is tested for bias towards heads: H0: p = 0.5, H1: p > 0.5, at the 5% significance level. For each number of tosses n, find the critical region and its actual probability when H0 is true.

    1. (a) n = 10
    2. (b) n = 20
    3. (c) n = 30

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