Hypothesis testing

• Jan 26th 2009, 07:03 AM
mirrormirror
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:

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

The producer decides to reject $\displaystyle H_0$ 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.

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

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

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?

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

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

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.
• Jan 26th 2009, 11:00 PM
Constatine11
Quote:

Originally Posted by mirrormirror
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:

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

The producer decides to reject $\displaystyle H_0$ 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.

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

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

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?

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

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

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 $\displaystyle z$ is negative you want $\displaystyle P(Z<z)=1-P(Z>|z|)$ for your probability, if that makes any sense?

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