# What is the exact relationship between confidence intervals and population parameters?

I was doing an online course, and the course seemed to suggest that the confidence interval says is that if we have a statement such as the following:

We are XX% confidence interval for some sample statistic is between $$\pm$$ y.

What this means is that if we take a sample from the population, we can be sure that XX% of the time, the true population statistic will be within: sample statistic $$\pm$$ y.

I am not sure why this is the case. I was under the impression that the confidence interval gave us a way to tell whether or not the sample statistic was within a certain interval of the true population mean.

Can someone pleas help me understand this?

• This and its linked threads may help. Jun 22, 2020 at 20:27

Given a sample you can find a confidence interval. The larger the sample, the narrower the interval. A different sample will give a different confidence interval, so that we generate a set of random intervals. For a 95% confidence interval we are saying that, 95% of the time, such a random interval will contain the true population parameter. It is all about the weight of evidence for the true parameter value, nothing is absolutely definitive.
Typically, you might claim the population parameter value has a certain value. Then if the confidence interval you calculate contains that value, you can say that you do not have evidence to reject the value claim. This is a bit weaker than saying the parameter value IS in the interval, but we can live with it.

• Hey, thanks! like I mentioned in my comment above, how do we know what the confidence interval is as per a given XX? Usually we use z-scores to determine the confidence interval, but that z-score is only pertinent to the current sample at hand right? How can we use it to make an estimate about the true population parameter when in reality the z-score is only telling us how variable our current sample is?
– user796478
Jun 22, 2020 at 13:58
• Calculating a conf. interval depends on the sample size, the parameter being estimated, what other parameter you might know (e.g. the variance) and the assumed underlying probability distribution for your sample. You willl need to be more specific with an example.
– Paul
Jun 22, 2020 at 15:01
• So 95% CI means 95% of the samples we take along with the generated CIs will contain the true population parameter right? So say we draw a sampling distribution of the means. We know that 95% of the obs fall within 2SE of this right? In that case, let's say we pick a random obs from this sampling distribution, how do we know whether this obs ± SE will contain the true population parameter? Couldn't it be so that this CI that we generate from the obs does not include the true population parameter because its CI is small even though the obs itself lied within 2SE of the sampling distributions?
– user796478
Jun 23, 2020 at 10:48
• Sorry, I can't make sense of what you are saying. You seem to be using the word obs to mean several observations and also one observation. You seem also to be talking about 2 confidence intervals, one with one SE and one with 2SE.
– Paul
Jun 23, 2020 at 16:36
• We have a population. We repeatedly take samples. For each sample, we create a 95% confidence interval. For another sample, we create another 95% confidence interval. Each time the associated CI will be a little different. How can we guarantee then that with the 95% CI, 95% of these samples will contain the true population parameter?
– user796478
Jun 24, 2020 at 7:08

The two statements you mention sound equivalent to me (if I understand what you are saying), but we are trying to estimate the true population statistic using a sample so you can think of the confidence interval as a margin of error of that estimate. However, the definition of confidence interval you give is incorrect. Once estimated using a sample, the conficende interval either contains or it doesn't contain the true population statistic. If you were to estimate it multiple times using different samples then XX% of those intervals would contain the true population statistic, but you don't know which ones.

• I understand this, but in that case how do we know what the confidence interval is as per a given XX? Usually we use z-scores to determine the confidence interval, but that z-score is only pertinent to the sample at hand right? How can we use it to make an estimate about the true population parameter?
– user796478
Jun 22, 2020 at 13:57
• The confidence interval is a random interval. With the sample at hand you calculate a realisation of the confidence interval, and use that to place error bars around the estimate of the statistic you are calculating, say the mean. Yes, the estimated CI is dependent on the sample but at least we know that (for a 95% CI) 95% of the intervals calculated this way will contain the true population mean. Jun 22, 2020 at 14:31
• Okay, but how do we know this? Our knowing this seems to come from the distribution of sample means.
– user796478
Jun 25, 2020 at 10:16