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    Multinomial, Negative Binomials and Hypothesis Testing

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    I have several questions I'm stuck on:

    1. A production line produces good articles with probability .7,
    average ones with probability of .2, and defective ones with
    probability .1. Ten articles are selected.

    a) What is the probability of 8 good ones and 1 defective?

    b) What is the probability that there is an equal number of good
    and defective articles?

    Answers in back of text: a) 0.1036 b) 0.006165

    I attempted to solve this using Multinomial Distribution.

    2. A person decides to throw a pair of dice until he gets 2 sixes.
    What is the expected number of throws until he stops?

    Answer in back of text: 36

    I attempted to solve this using Negative Binomial Distribution.

    3. A transistor manufacturer claims its product has 10% defectives.
    A sample of 15 transistors is examined and 3 are found to be
    defective. Would you reject their claim with alpha <= 0.05?

    Answer in back of text: No

    4. An experiment is set to test the hypothesis that a given coin is
    unbiased. The decision rule is the following: Accept the hypothesis
    if the number of heads in a sample of 200 tosses is between 90 and
    110 inclusive, otherwise reject the hypothesis.

    a) Find the probability of accepting the hypothesis when it is
    correct.

    b) Find the probability of rejecting the hypothesis when it is
    actually correct.

    Answer in back of text: a) 0.8612 b) 0.1388

    Thanks in advance.

    © BrainMass Inc. brainmass.com December 24, 2021, 5:12 pm ad1c9bdddf
    https://brainmass.com/statistics/hypothesis-testing/multinomial-negative-binomials-hypothesis-testing-34579

    SOLUTION This solution is FREE courtesy of BrainMass!

    Please refer to response file attached (also see below). I hope this helps and take care.

    RESPONSE:

    Statistics: Multinomial, Negative Binomial, Hypothesis Test

    1. a) What is the probability of 8 good ones and 1 defective?

    10!
    P(8 good, 1 av, 1 defective) = -------- x 0.7^8 x 0.2 x 0.1
    8! 1! 1!

    = 90 x 0.7^8 x 0.2 x 0.1

    = 0.103766

    b) What is the probability that there is an equal number of good
    and defective articles?

    Work out the following probabilities (in order good, average,
    defective):

    0, 10, 0
    1, 8, 1
    2, 6, 2
    3, 4, 3
    4, 2, 4
    5, 0, 5

    2. A person decides to throw a pair of dice until he gets 2 sixes.
    What is the expected number of throws until he stops?

    You can use a difference equation.

    Let E = expected number of throws to a double 6

    You MUST throw at least once and there is 35/36 probability of
    returning to the start point:

    E = 1 + (35/36)E

    E(1 - 35/36) = 1

    E(1/36) = 1

    E = 36

    So the expected number of throws is 36.

    3. A transistor manufacturer claims its product has 10% defectives.
    A sample of 15 transistors is examined and 3 are found to be
    defective. Would you reject their claim with alpha <= 0.05?

    Proportion defective = 1/5 = 0.2 while claim is 0.1

    p = 0.1

    pq/n = 0.1 x 0.9/15 = 0.006

    sqrt(pq/n) = 0.07746

    We test

    0.2 - 0.1
    z = --------- = 1.291
    0.07746

    and we compare this with 1.645 (single tailed test), and see that it
    is not significant. So do not reject null hypothesis.

    4. a) Find the probability of accepting the hypothesis when it is
    correct.

    p=q = 1/2

    npq = (1/2)(1/2)(200) = 50

    sqrt(npq) = 7.071

    110 - 100 10
    z = --------- = ----- = 1.4142
    7.071 7.071

    A(z) = 0.9214

    so area from mean to this value is 0.4214. By symmetry area other side
    of the mean is also 0.4214. Total area corresponding to range 90 - 110
    is 0.8428, so we can accept null hypothesis with probability 0.8428.

    b) Find the probability of rejecting the hypothesis when it is
    actually correct.

    This is simply 1 - the previous answer.

    1 - 0.8428 = 0.1572

    BEST OF LUCK!

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 5:12 pm ad1c9bdddf>
    https://brainmass.com/statistics/hypothesis-testing/multinomial-negative-binomials-hypothesis-testing-34579

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