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Math Help - Hypothesis testing

  1. #1
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    Hypothesis testing

    The producer of an electric machine wants to make a test to see how many of their machines that are working correct. The hypothesis are the following:

    H_0: the proportion of defect machines is at most 8%
    H_a: the true proportion of defect machines is more than 8%

    The producer decides to reject H_0 if the sample proportion of defect machines are more than 12%.

    i) If we draw a sample of 100 machines, then what is the probability of making a type I error?

    I'm thinking of a Z-transformation.

    P(type 1)=P(phat{>0.12}\mid p=0.08) = P(Z>z)

    z = (0.12-0.08)/\sqrt{0.08(0.92)/100)}

    And then I look up P(Z>z) in a table. Is that correct?

    ii) If we know that the true proportion of defective machines is 15%. Then what is the probability of a type II error with a random sample of 100?

    P(type 2)=P(phat{<0.12}\mid p=0.15) = P(Z>z)

    z = (0.12-0.15)/\sqrt{0.15(0.85)/100)}

    Since this will be negative, I look up the probability that Z(<-z) in a table. Do I think in the right way?

    Thanks a lot for your help.
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  2. #2
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    Quote Originally Posted by mirrormirror View Post
    The producer of an electric machine wants to make a test to see how many of their machines that are working correct. The hypothesis are the following:

    H_0: the proportion of defect machines is at most 8%
    H_a: the true proportion of defect machines is more than 8%

    The producer decides to reject H_0 if the sample proportion of defect machines are more than 12%.

    i) If we draw a sample of 100 machines, then what is the probability of making a type I error?

    I'm thinking of a Z-transformation.

    P(type 1)=P(phat{>0.12}\mid p=0.08) = P(Z>z)

    z = (0.12-0.08)/\sqrt{0.08(0.92)/100)}

    And then I look up P(Z>z) in a table. Is that correct?

    ii) If we know that the true proportion of defective machines is 15%. Then what is the probability of a type II error with a random sample of 100?

    P(type 2)=P(phat{<0.12}\mid p=0.15) = P(Z>z)

    z = (0.12-0.15)/\sqrt{0.15(0.85)/100)}

    Since this will be negative, I look up the probability that Z(<-z) in a table. Do I think in the right way?

    Thanks a lot for your help.
    this looks OK, for the second case as z is negative you want P(Z<z)=1-P(Z>|z|) for your probability, if that makes any sense?

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