Critical value (z*) for a given confidence level | AP Statistics | Khan Academy

We're told that Elena wants to build a one sample z interval to estimate what proportion of computers produced at a factory have a certain defect. She chooses a confidence level of 94%. A random sample of 200 computers shows that 12 computers have the defect. What critical value z star should Elena use to construct this confidence interval?

So before I even ask you to pause this video, let me just give you a little reminder of what a critical value is. Remember, the whole point behind confidence intervals is that we have some true population parameter; in this case, it is the proportion of computers that have a defect.

So there's some true population proportion we don't know what that is, but we try to estimate it. We take a sample—in this case, it's a sample, a random sample of 200 computers. We take a random sample, and then we estimate this by calculating the sample proportion.

But then we also want to construct a confidence interval. Remember, a confidence interval at a 94% confidence level means that if we were to keep doing this and if we were to keep creating intervals around these statistics, so maybe that's the confidence interval around that one, maybe if we were to do it again, that's the confidence interval around that one.

That 94%, roughly as I keep doing this over and over again, that 94% of these intervals are going to overlap with our true population parameter. The way that we do this is we take the statistic—let me just write this in general form. Even if we're not talking about a proportion, it could be if we're trying to estimate the population mean.

So, we take our statistic, and then we go plus or minus around that statistic. Plus or minus around that statistic, and then we say, okay, how many standard deviations for the sampling distribution do we want to go above or beyond? So, the number of standard deviations we want to go, that is our critical value.

Then we multiply that times the standard deviation of the statistic. Now, in this particular situation, our statistic is p-hat from this one sample that Elena made. So, the one sample proportion that she was able to calculate plus or minus z star.

And we're going to think about which z star, because that's essentially the question: the critical value. So, plus or minus some critical value times—and what we do, because in order to actually calculate the true standard deviation of the sampling distribution of the sample proportions, well then you actually have to know the population parameter. But we don't know that, so we multiply that times the standard error of the statistic.

We've done this in previous videos, but the key question here is: what is our z star? What we really need to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be a true population parameter, which we do not know.

But how many standard deviations above and below the mean in order to capture 94% of the probability? 94% of the area—so, this distance right over here where this is 94, this number of standard deviations, that is z star right over here.

Now, all we have to really do is look it up on a z table. But even there, we have to be careful, and you should always be careful which type of z table you're using, or if using a calculator function, what your calculator function does. Because a lot of z tables will actually do something like this.

For a given z, they'll say what is the total area going all the way from negative infinity up to including z standard deviations above the mean? So when you look up a lot of z tables, they will give you this area.

So one way to think about this is, we want to find the critical value. We want to find the z that leaves not 6% unshaded in, but leaves 3% unshaded in. Where did I get that from? Well, 100% minus 94% is 6%, but remember, this is going to be symmetric on the left and the right.

So you're gonna want 3% not shaded in over here and 3% not shaded in over here. So when I look at a traditional z table that is viewing it from this point of view, this cumulative area, what I really want to do is find the z that is leaving 3% open over here, which would mean the z that is filling in 97% over here—not 94%, but if I find this z.

But if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%. So that out of the way, let's look that up. What z gives us fills us fills in 97% of the area?

So, I got a z table—this is actually the one that you would see if you’re, say, taking AP statistics—and we would just look up where do we get to 97%. And so it is, 97 looks like it is right about here. That looks like the closest number. This is six ten-thousands above it; this is only one ten-thousandth below it, and so this is, let's see.

If we look at the row, it is 1.88. So, going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area up to including 1.88 standard deviations above the mean, that would be 97%.

But if you were to go 1.88 standard deviations above the mean and 1.88 standard deviations below the mean, that would leave 3% open on either side, and so this would be 94%. So, this would be 94%.

But to answer their question, what critical z value or what critical value z star? Well, this is going to be 1.88, and we're done.