If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 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.
• How do you get 97% from 94%?
• 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
• 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?
• 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).
• 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?
• 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!
• 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?
• 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)
(1 vote)
• What's the method for finding this with a TI-84 calculator?
• 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.