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Critical value (z*) for a given confidence level

If we want to be 95% confident, we need to build a confidence interval that extends about 2 standard errors above and below our estimate. More precisely, it's actually 1.96 standard errors. This is called a critical value (z*). We can calculate a critical value z* for any given confidence level using normal distribution calculations.

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Video transcript

- [Instructor] 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 are 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 wanna construct a confidence interval. And 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% that roughly as I keep doing this over and over again, that 90, that roughly 94% of these intervals are going to overlap with our true population parameter. And 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, 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 wanna go above or beyond? So the number of standard deviations we wanna go, that is our critical value, and then we multiply that times the standard deviation of the statistic, of the statistic. Now in this particular situation, our statistic is p hat from this one sample that Elena made. So it's that one sample proportion that she was able to calculate, plus or minus z star, and we're gonna 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. And we've done this in previous videos. But the key question here is, what is our z star? And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it which would actually be our 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 you're 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, above the mean? So when you look up a lot of z-tables, they will give you, they will give you this area. So one way to think about this, we wanna find the critical value, we wanna 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 wanna 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 were 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 .0006 above it. This is only .0001 below it. And so this is, let's see you would look at the row first. If we look at the row it is 1.8, 1.88 is our z. 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 and 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 is z star? Well this is going to be 1.88, and we're done.