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# Critical value (z*) for a given confidence level

AP.STATS:
UNC‑4 (EU)
,
UNC‑4.D (LO)
,
UNC‑4.D.1 (EK)
,
UNC‑4.D.2 (EK)

## Video transcript

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 percent a random sample of 200 computers show 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 the 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 and remember a confidence interval it would not what at a 94 percent 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 percent that roughly as I keep doing this over and over again that 90 that roughly 94 percent 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 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 want to go above or beyond so the number of standard deviations we want to 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 said one sample proportion that she was able to calculate plus or minus Z star and we're gonna think about which is Z star cuz that's essentially the question the critical value so plus or minus some critical value x 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 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 a true population parameter which we do not know but how many standard deviations above and below the mean in order to capture 94 percent of the probability 94 percent of the area so this distance right over here where this is 94 percent 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 given Z they'll say what is the total area going all the way Nate 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 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 a hundred percent minus 94 percent is 6% but remember this is going to be symmetric on the left on the right so you're going to want three percent not shaded in over here and three percent not shade didn't 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 percent over here not 94 percent but if I find this Z but if I were to stop it right over there as well then I would have three percent available there and then the true area that we're filling in would be 94 percent so with that out of the way let's look that up what Z gives us fills us fills in 97 percent of the area so I got AZ 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 percent and so it is 97 percent looks like it is right about here that looks like the closest number this is 6 10 thousandths above it this is only one ten-thousandth 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 1.8 8 is our Z so going back to this right over here if our Z is equal to one point eight eight so this is equal to one point eight eight then all of this area up to an including one point eight eight standard deviations above the mean that would be 97 percent but if you were to go one point eight eight standard deviations above the mean and one point eight eight standard deviations below the mean that would leave three percent open on either side and so this would be 94 percent so this would be 94 percent but to answer their question what critical z-value or what critical values Z star well this is going to be one point eight eight and we're done
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