type 1 and 2 error easy help please

#1
Lifetime of a certain type of batteries is said to be Normally distributed with a known standard deviation of 5 hours. The manufacturer claims that the average lifetime of the batteries is 125 hours. A sample of size 100 is taken from the population. An investigator wishes to test the hypotheses Ho: mu = 125 against H1: mu <125. He decides on the following criteria:
Accept Ho if the sample mean xbar is ≥ 124
Reject Ho if the sample mean xbar is < 124

(i)Find the probability of making a Type I error.

(ii)If the true mean is 123.75 hours, find the probability of Type II error and hence the power of the test.

(iii)Suppose that the investigator wants the probability of Type I error fixed at α=0.05. Revise his decision criteria by finding a value of the sample mean below which he should reject Ho. Also calculate the power of this test when the true mean is 123.75 hours.



Now, i have got part (i) answer as 0.0228. However i am not sure how to go about doing (ii). I have ideas in my head that i am finding hard to visualise. Part (iii) i can manage i think, although guidance would be awesome :)
 

Dason

Ambassador to the humans
#2
For part II why don't you tell us what your ideas are. Do you know what a type II error is? Do you know what power is in the context of testing?
 
#3
yes i do know what type 2 error is. its just finding Ho to be true, when in fact it is false. Power of the test is just 1-B where B is the probability of type 2 error
 
#4
i know definition by heart, and i understand what it means and the point of type 1 and 2 error and what it does. But when posed with questions that are a bit different, i just get stuck.

By the way, please tell me part (i) is right or i might have to start learning probability all over again
 

Dason

Ambassador to the humans
#5
So for part II if the mean is actually 123.75 then Ho isn't true. So what you want to do is find the probability that the test statistic isn't in the rejection region (in this case the rejection region is any value below 124) given that the mean is actually 123.75.
 
#8
i tried findin xbar first, then putting that back to the z score

xbar = Z*S.d+mu
xbar = -1.96(<--is this right?) *0.5+125
xbar = 124.05

put that into z score

z= (124.02-123.75)/0.5

z= 0.54

from z table p(type 2 error)=0.7054


that's what i got, i am not sure about my initial z crit, and i am tired of this question. Watched too many vids from khanacademy. Anyways, i am off to sleep. I will look at what you've written tommorow morning. Thanks!
 
#9
ok, so i think the -1.96 is probably wrong. Since i got the Z-crit as -2 in question (i), thats probably what i should use.

z=0.5
which gives me p(type 2 error)=0.6915

if anyone could verify this, would be appreciated