Probability of Type I and II errors when P = a number

oxyron · #1 $Old$ December 3rd, 2008, 02:15 PM

I have the following problem:
"When a thread-cutting machine is operating properly, only 2% of the units produced are defective. Since the machine was bumped by a forklift, the quality-control manager is concerned that it may require expensive downtime and readjustment. The manager wishes to test, null hypothesis H0: P =< 0.02 against H1: P > 0.02. Using a random sample of 400 units, his decision rule for the test is defined as: Reject H0 if P-hat > 0.034; and don't reject H0 if P-hat =< 0.034."
a) Find probability of Type I error (alpha) when P = 0.02
b) Find probability of Type II error (beta) and the power, when P = 0.04
c) Find power of test when the true proportion P = 0.05

My issue is, we just started this material (what H0 and H1 is, and the two types of errors) and have not covered this yet. So, I went to the book to gain a clue. I assume since P is =< 0.02, we will want to use a right tail/upper tail test. However, I'm not sure if we use Z or t, and really, where I go from here to answer a b and c.

Please help?

mr fantastic · #2 $Old$ December 4th, 2008, 03:58 AM

Quote:

Originally Posted by oxyron $View Post$

I have the following problem:
"When a thread-cutting machine is operating properly, only 2% of the units produced are defective. Since the machine was bumped by a forklift, the quality-control manager is concerned that it may require expensive downtime and readjustment. The manager wishes to test, null hypothesis H0: P =< 0.02 against H1: P > 0.02. Using a random sample of 400 units, his decision rule for the test is defined as: Reject H0 if P-hat > 0.034; and don't reject H0 if P-hat =< 0.034."
a) Find probability of Type I error (alpha) when P = 0.02
b) Find probability of Type II error (beta) and the power, when P = 0.04
c) Find power of test when the true proportion P = 0.05

My issue is, we just started this material (what H0 and H1 is, and the two types of errors) and have not covered this yet. Mr F says: Then wouldn't it be better then to wait until it has been covered ....?

So, I went to the book to gain a clue. I assume since P is =< 0.02, we will want to use a right tail/upper tail test. However, I'm not sure if we use Z or t, and really, where I go from here to answer a b and c.

Please help?

(0.034)(400) = 13.6.

a) Calculate Pr(X > 13 | X ~ Binomial(p = 0.02, n = 400)).

b) Calculate Pr(X < 14 | X ~ Binomial(p = 0.04, n = 400)).

Power $= 1 - \beta$ .

oxyron · #3 $Old$ December 4th, 2008, 05:29 AM

Quote:

Mr F says: Then wouldn't it be better then to wait until it has been covered ....?

I agree 100% with you. However the professor gave it out on Tuesday and it is due today (Thursday), so I don't have much choice in what we cover or when it needs to be turned in. Thanks for the assistance, by the way.