How to find a confidence level given the z value

Question

I have a task from which I would like to find the confidence level given the z value. I have a sample population. From that population, given its distribution, I would like to find the confidence level of a given value of that population. In other words, given a a value of the population, I would like to know if it is within 95% (confidence level) of the whole population or 68% or 50% and so on. Usually, we can find the z value and confidence interval and given confidence level as explained here How to find the confidence Interval. But I would like to find the confidence level given the z value (which in this case is a given value from the population).

How can I tackle this? If possible it should be in python or in R

Answer 1

OK, for a 95% confidence interval, you want to know how many standard deviations away from the mean your point estimate is (the "z-score"). To get that, you take off the 5% "tails". Working in percentile form you have 100-95 which yields a value of 5, or 0.05 in decimal form.

Divide that in half to get 0.025 and then, in R, use the qnorm function to get the z-star ("critical value"). Since you only care about one "side" of the curve (the values on either side are mirror images of each other) and you want a positive number, pass the argument lower.tail=FALSE.

So, in the end, it would look like this:

qnorm(.025,lower.tail=FALSE)

yielding a value of 1.959964

You then plug that value into the equation for the margin of error to finish things up.

If you want to go the other direction, from a "critical value" to a probability, use the pnorm function. Something like:

pnorm(1.959964,lower.tail=FALSE)

which will give you back 0.025

Answer 2

To convert between z-scores and confidence values with python, use the cdf and ppf functions in scipy.stats.norm.

There is a nice example of how to use them in the answer for this question.