What does the (1-alpha) mean in confidence interval?

Question

same as title, what does it represents? I know alpha is the percentile we are trying to reach. such as if we are trying to get to 95% CI , then 0.05 would be 1-alpha .

the question is :explain what is meant by the“1-ɑ”part of a confidence interval

Answer 1

Let's say you want to estimate what the mean of a distribution is with $95$ percent certainty. You have a good number of observations (i.e., $n$ is "large"); you have a sample mean, you know the population variance. Then you set up the confidence interval for the mean like so:

$$\left[\bar{x} - z_{\alpha / 2} \left(\frac{\sigma}{\sqrt{n}}\right),\, \bar{x} +z_{\alpha / 2} \left( \frac{\sigma}{\sqrt{n}} \right) \right]$$

which, for $\alpha = 0.05$, is

$$\left[\bar{x} - 1.96 \left(\frac{\sigma}{\sqrt{n}}\right),\, \bar{x} + 1.96 \left( \frac{\sigma}{\sqrt{n}} \right) \right]$$

The central limit theorem tells us that, given enough observations, the random variable $W = \frac{\bar{X}-\mu}{\sigma / \sqrt{n}}$ has a standard normal distribution that is, it has a p.d.f. for which we can draw a curve. If we were to find the area under that curve between the endpoints of our interval above, we'd find that it equals $0.95$ in other words, $1-\alpha$. Since $W$ is the distribution of our sample means, the population mean could theoretically live anywhere underneath the curve for the p.d.f.; however, most of the area under the curve is near the center of the distribution. Because of this fact, and because the area under the entire curve is $1$, we can take a relatively small set of the possible values $\bar{x}$ might take on and say that $95$ percent of the time $\mu$ will be within that set. (I've described a two-tail test, but the idea is same for one tail.)

Put another way, you can say with $100$ percent confidence that $\mu$ is in the interval $(-\infty,\infty)$. If you want to narrow that interval a bit, though, you'll have to sacrifice some certainty. If you shave off a tail or two corresponding to an area of $\alpha$, you'll have $100(1-\alpha)$ percent confidence.

Answer 2

It's much simpler than that... the $1$ simply means a whole number. So, an alpha of $0.05$ tells you that you're looking for a $95\%$ confidence interval... ergo, $1 - 0.05 = 0.95$. That can be calculated with $100\%$ confidence.