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IB Mathematics: Applications and Interpretation HL · 13.4: Hypothesis Testing · 13.4.21

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Exam question practice Problem set B

Show each probability as a calculator step. Name the distribution before using it.

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. The mass, X kilograms, of a pumpkin is normally distributed with mean 3.4 and standard deviation 0.6. Find the probability that

    1. (a) one pumpkin has a mass greater than 4 kg;
    2. (b) two pumpkins chosen at random both have a mass greater than 4 kg.

  2. The mass, X kilograms, of a pumpkin is normally distributed with mean 3.4 and standard deviation 0.6. Two pumpkins are chosen at random. Find the probability that their total mass is greater than 7.5 kg.

  3. A grower uses a new fertiliser, which increases the mass of every pumpkin by 5%. Before the change the mass has mean 3.4 kg and standard deviation 0.6 kg.

    1. (a) Find the new mean.
    2. (b) Find the new standard deviation.

In context

  1. A grower thinks that a new fertiliser has increased the mean mass of a pumpkin. The old mean was 3.4 kg and the standard deviation is 0.6 kg. He tests a random sample of 20 pumpkins at the 5% significance level.

    1. (a) State the null and alternative hypotheses.
    2. (b) Find the critical region for the sample mean. Give the critical value to 3 significant figures.

  2. The new fertiliser in fact increases the mass of every pumpkin by 5%. A test of H0: μ = 3.4 against H1: μ > 3.4 uses a sample of 20 pumpkins, a standard deviation of 0.6 kg and a 5% significance level. Before the change the mean mass is 3.4 kg.

    1. (a) Find the probability of a Type II error. Use the unrounded critical value.

Stretch

  1. The mass, X kilograms, of a pumpkin is normally distributed with mean 3.4 and standard deviation 0.6. A grower uses a new fertiliser and thinks the mean mass has increased. He tests this at the 5% significance level, using a random sample of 20 pumpkins.

    1. (a) Find the probability that a pumpkin has a mass greater than 4.0 kg.
    2. (b) Two pumpkins are chosen at random. Find the probability that their total mass is greater than 7.5 kg.
    3. (c) State the null and alternative hypotheses.
    4. (d) Find the critical region for the sample mean.
    5. (e) The fertiliser in fact makes every pumpkin 5% heavier. Find the new mean and standard deviation of the mass of a pumpkin.
    6. (f) Find the probability of a Type II error. Use the unrounded critical value.

jmaths.xyz · AI HL · 13.4.21 Exam question: sums of normals, a test for a mean and Type II error