00:01
So we have a manufacturer is interested in the output voltage of a power supply used in a pc and the we're told it's the output voltage is normally distributed and that's why this normal curve is here and the same duration is 0 .2 volts and the manufacturer wants to test h not that the mean is 4 volts and the alternative is that the mean is not 4 volts and we're gonna find what we're trying to do today is find the probability of a type 2 error if the true mean output is mu sub t is equal to 0 4 .1 and we're gonna check this out the alpha 0 .01 level of significance and we're told the sample size is going to be 20 so we can use our z distribution because we're told it's normally distributed and we have the population standard deviation sigma 0 .2 so we're good there.
01:03
So this is our normal distribution so what we're gonna do to where you find that the type 2 error is you you don't look at just the one you have to compare both together.
01:14
So that's what this extra piece is down here this the green would be the one where the mean is 4 is where your mean would be 4 where the red will be the? where the mean is 4 .1.
01:27
So down here.
01:28
This is where the mean is 4 .4.
01:30
Here is 4 .1 so what you do and i'm gonna do this have a separate one here so we can just kind of compare so if we're looking at the initial test this one we'd have a crit some critical value so here's our mean of 4 we have some lower bound here some x bar value lower value here and then some x our upper value here and right here this black area shading this if you add these two areas together you get that alpha 0 .01 and the type 2 error if we go down here this is not to scale this is just to kind of show you so again you have you have your critical values here and here for the green you don't make these green so we can this is the green.
02:22
This is the green critical values based on x bar and then x bar upper so this is the cutoff here.
02:34
The green areas are the alpha put them together get the alpha the type 2 error is here it's where you take those critical values and you look at the area under the new red curve between those critical values those those values here and so this is what we get so we're looking for this pink area this magenta area this this is our type 2 error it's also called beta and that is the type 2 error because what the type 2 error is it's the probability of failing to reject when you should right that's because if you'd fail to reject if your if your values fall in this region here so we need to find this value so what we're gonna do is we're gonna find these critical values here for our give against this initial hypothesis and then we're gonna find the probability being between them so we're gonna convert we're gonna find these lower values here by using the z transform x bar so as these have x bar is equal to x bar minus the mu mu sub x bar which is gonna be 4 divided by sigma over the square root of n so we can do this can we get rid of this so our what we're gonna do to get our z scores we're gonna take each of the x bar.
03:59
We don't need know the x bar yet, but we need the z score so the z score that corresponds with zero point zero one over two is two point five eight it's x bar minus four divided by this sigma over root n that is the standard deviation 0 .2 divided by square root of 20 and that's called the standard error...