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Math Help - Type 1/ Type 2 Error Question

  1. #1
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    Type 1/ Type 2 Error Question

    I have problems trying to construct a proper framework to deal with this one. Please help if you can!

    A random sample of 500 registered voters in a rural city is asked if they favor abolition of road tax. If more than 400 voters respond positively, we will conclude that at least 60% of the voters favor the abolition of road tax.
    a) Find the probability of Type 1 error if exactly 60% of the voters favor the abolition of road tax.
    b) What is the type II error probability if 75% of the voters favor the abolition of road tax?

    Thanks!
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  2. #2
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    Quote Originally Posted by chopet View Post
    I have problems trying to construct a proper framework to deal with this one. Please help if you can!

    A random sample of 500 registered voters in a rural city is asked if they favor abolition of road tax. If more than 400 voters respond positively, we will conclude that at least 60% of the voters favor the abolition of road tax.
    a) Find the probability of Type 1 error if exactly 60% of the voters favor the abolition of road tax.
    b) What is the type II error probability if 75% of the voters favor the abolition of road tax?

    Thanks!
    H0: Less than 60% of the voters favor the abolition of road tax.

    H1: More than 60% of the voters favor the abolition of road tax.

    Let \hat{p} represent the proportion of the sample that favor the abolition of road tax.

    Then (for large n) \hat{p} ~ Normal \left( \mu = p, ~ \sigma = \sqrt{\frac{p(1-p)}{n}}\right)

    where p is the proportion of the population that favor the abolition of road tax and n is the sample size.



    Probability of type I error = Pr(Reject H0 | H0 true) = \Pr(\hat{p} > 0.8 | p = 0.6)

     = \Pr(\hat{p} > 0.8 | \hat{p} ~ Normal \left( \mu = 0.6, ~ \sigma = \sqrt{\frac{0.6(1-0.6)}{500}} = 0.022\right).



    Probability of type II error = Pr(Don't reject H0 | H0 false) = \Pr(\hat{p} < 0.8 | p = 0.75)

     = \Pr( \hat{p} < 0.8 | \hat{p} ~ Normal \left( \mu = 0.75, ~ \sigma = \sqrt{\frac{0.75(1-0.75)}{500}} = 0.019\right).


    Calculations are left for you.
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  3. #3
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    Thanks for the fast response.
    Just to clarify:

    For the Type 1 error:
    All I have to do is calculate the z-value of 0.8 with a mean of 0.6:
    {{0.8-0.6} \over {0.022} } = 9.09

    Won't that be way off the normal curve. Hence probability of Type 1 error is 0?
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  4. #4
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    Quote Originally Posted by chopet View Post
    Thanks for the fast response.
    Just to clarify:

    For the Type 1 error:
    All I have to do is calculate the z-value of 0.8 with a mean of 0.6:
    {{0.8-0.6} \over {0.022} } = 9.09

    Won't that be way off the normal curve. Hence probability of Type 1 error is 0?
    Yes.
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  5. #5
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    this values is true???

    let x1,x2......xn be a random samole from N(mu,9) it is decided that the probability of rejecting :
    mu=1 when mu=1.8 is (0.5)

    and the probability of accepting mu<or equal 1 when mu=1.2 is (0.1)


    if the test use :
    mean(x bar)>K

    find n and K:

    solution :

    i find this two unknowns and i get :

    n= 304

    and K=.799
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