Book 1: Question regarding type I/II error

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In Schweser Book1 page 274:

25) For a hypothesis test with a probability of a Type II error of 60% and a proabaility of a Type I error of 5%, which of the following is TRUE?

a) Power of test is 40%, 5% probability that the test statistic will exceed the critial value(s)
b) There is a 95% probability that the test statistic will be between the critical values if this is a two-tail test
c) definately wrong answer
d)There is a 5% probability that the null hypothesis will be rejected when actually true, and the probability of rejcing the null when it is false is 40%

The correct answer is D and mentions "A and B are not necessarily ture, since the null may be false and the probability of rejection unknown."

Why is the probability of rejection unknown? I thought the probability of rejection is based on alpha (level of significance)

Any help on this would be very much appreciated!
 
Chance of a type 1 error = the level of significance, alpha. Usually the type 2 error probability= Beta = (1-Alpha), but in this case it is stated at 60% = Beta, a type two error occurs when you accept a false hypothesis, therefore the probability of rejecting a null hypothesis when it is false (which is the right decision) is (1-Beta) or 40%.
 
jamespucyk Wrote:
-------------------------------------------------------
> Chance of a type 1 error = the level of
> significance, alpha. Usually the type 2 error
> probability= Beta = (1-Alpha), but in this case
> it is stated at 60% = Beta, a type two error
> occurs when you accept a false hypothesis,
> therefore the probability of rejecting a null
> hypothesis when it is false (which is the right
> decision) is (1-Beta) or 40%.


James, you are correct but I was referring to the phrase "since the null may be false and the probability of rejection unknown." Why is this true? I thought the probability of rejection is based on alpha which is stated or "given".
 
Think about it like this, with a 60% chance of a type 2 and 5% of a type you have a range of probabilities where the null will definatly be accepted if true, namely 95% (1-alpha) - beta (60%), so there is a 35% window, around the sample mean. So think about it like the bell curve, 35% around the mean where you don't have to worry about making an error, 60% toward the tail where there is a possibility that you will accept a false null and a 5% chance of rejecting a true null, as you reduce alpha beta will increase prorportionally, but the 35% zone around the mean is a safe zone.

Put another way if you find the T-stat falls in that 35% zone, your cool.
 
jamespucyk Wrote:
-------------------------------------------------------
> Think about it like this, with a 60% chance of a
> type 2 and 5% of a type you have a range of
> probabilities where the null will definatly be
> accepted if true, namely 95% (1-alpha) - beta
> (60%), so there is a 35% window, around the sample
> mean. So think about it like the bell curve, 35%
> around the mean where you don't have to worry
> about making an error, 60% toward the tail where
> there is a possibility that you will accept a
> false null and a 5% chance of rejecting a true
> null, as you reduce alpha beta will increase
> prorportionally, but the 35% zone around the mean
> is a safe zone.
>
> Put another way if you find the T-stat falls in
> that 35% zone, your cool.

Thank you James. Your explanation was thorough and clear.
You are an asset to the forums :)
 
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