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### Course: AP®︎/College Statistics > Unit 10

Lesson 2: Confidence intervals for proportions- Conditions for valid confidence intervals for a proportion
- Conditions for confidence interval for a proportion worked examples
- Reference: Conditions for inference on a proportion
- Conditions for a z interval for a proportion
- Critical value (z*) for a given confidence level
- Finding the critical value z* for a desired confidence level
- Example constructing and interpreting a confidence interval for p
- Calculating a z interval for a proportion
- Interpreting a z interval for a proportion
- Determining sample size based on confidence and margin of error
- Sample size and margin of error in a z interval for p

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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.

## Want to join the conversation?

- I think Sal missed a video about explaining the basics of calculating confidence intervals? He said he was reminding us about a critical value when we never went over it. Can someone please tell me where that video is if he made it. Thanks.(12 votes)
- Here's the video about confidence intervals:

https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/introduction-to-confidence-intervals/v/confidence-intervals-and-margin-of-error(3 votes)

- How do you get 97% from 94%?(7 votes)
- in this lecture, the question asked 94%. I understood why we wanted 97%. Since its the cumulative probability.Leaving 6% to the right. We wanted an interval equidistant from the mean above/below. I just feel after we calculated the z-score. we need to subtract something like 3%. Like now, we getting the interval and the 3% to the left. Where did I go wrong(5 votes)
- You're correct in your understanding. The 97% probability is used to find the z-score that corresponds to the desired confidence level, which is 94%. The reasoning behind looking for 97% instead of 94% is to ensure symmetry in the confidence interval. The 3% on each side of the distribution ensures that the confidence interval is centered around the mean. After calculating the z-score for 97% probability, you would indeed subtract 3% from each tail to obtain the 94% confidence interval. So, you're not wrong in your understanding.(1 vote)

- I know how to get z* but how would you do the reverse, using a z* to get a confidence level?(4 votes)
- It helped me to write out all of the steps from CI% to z* and follow it in reverse.

Recall that CI%'s are symmetrical, so the area*outside*the CI% = 2*(1-p).(2 votes)

- why do we multiply z by standard deviation?

i imagine that

z=(p_hat-p)/standardDeviation

then

p_hat-p= z*standardDeviation

would that mean that the marginal error (z*stdDev) represent the difference between the populatin proportion and the sample proportion?(3 votes)- The conceptual meaning of z is the number of standard deviations from the mean. So the amount of difference from the mean (in this case, p_hat-p) is z times the standard deviation.

Have a blessed, wonderful day!(2 votes)

- Say you have 10 confidence intervals. Would averaging the first milestones and averaging the second milestones and making one confidence interval from that be valid?(2 votes)
- I think it is not valid. Let say that the margin of error is still correct but in this case, the new interval will depends on the first and the second so the it will be biased.(1 vote)

- I can't find the z-table, please help!(1 vote)
- What's the method for finding this with a TI-84 calculator?(7 votes)
- I don't know TI-84 has an inverse of error function or not. But if it has you can calculate by using sqrt(2)*inverf(x) where inverf is an inverse of error function and x is the confidence level (0.94 in this case)(1 vote)

- So, if Pp is the population proportion and Sp is the sample proportion, we have Pp = Sp +/- (1.88 * Standard Deviation).

Is Sp = 12 (defective computers)?(1 vote)- Yes, in the formula for constructing a confidence interval for a population proportion (p), the sample proportion (p-hat) is used. In this case, Sp represents the sample proportion of defective computers, which is 12 out of the 200 computers sampled. So, Sp = 12/200 = 0.06.(2 votes)

- Is there a way to calculate z* on a normal calculator? I think I'm missing something obvious!(1 vote)
- For scientific calculators, you can calculate the confidence level using the normalcdf function (the lower and upper boundaries will be negative and positive z*, respectively). You can also find z* by using the Inverse Normal (invNorm) function. Try searching your calculator on the Internet to check if you have these functions.

There are also plenty of online z* calculators if you need one.

Hope this helps!(2 votes)

## 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.