Confidence Intervals咩野離架?

A confidence interval (CI) for a population parameter is an interval with an associated probability p that is generated from a random sample of an underlying population such that if the sampling was repeated numerous times and the confidence interval recalculated from each sample according to the same method, a proportion p of the confidence intervals would contain the population parameter in question.

Practical example
A machine fills cups with margarine, and is supposed to be adjusted so that the mean content of the cups is close to 250 grams of margarine. Of course it is not possible to fill every cup with exactly 250 grams of margarine. Hence the weight of the filling can be considered to be a random variable X. The distribution of X is assumed here to be a normal distribution with unknown expectation μ and (for the sake of simplicity) known standard deviation σ = 2.5 grams. To check if the machine is adequately adjusted, a sample of n = 25 cups of margarine is chosen at random and the cups weighed. The weights of margarine are
圖片參考：http://upload.wikimedia.org/math/c/9/1/c916ca889ccfcef81cf716f93fdcc3d6.png
, a random sample from X.
To get an impression of the expectation μ, it is sufficient to give an estimate. The appropriate estimator is the sample mean:


圖片參考：http://upload.wikimedia.org/math/0/2/d/02d0a2c30e2af8ba2660821c2f1ff92c.png

The sample shows actual weights
圖片參考：http://upload.wikimedia.org/math/8/6/2/8621d1f417ca7380dec293acfea97e72.png
, with mean:


圖片參考：http://upload.wikimedia.org/math/5/2/5/5250e6b05ffd0714dc58e3fe6d38dbe5.png
(grams).
If we take another sample of 25 cups, we could easily expect to find values like 250.4 or 251.1 grams. A sample mean value of 280 grams however would be extremely rare if the mean content of the cups is in fact close to 250g. There is a whole interval around the observed value 250.2 of the sample mean within which, if the whole population mean actually takes a value in this range, the observed data would not be considered particularly unusual. Such an interval is called a confidence interval for the parameter μ. How do we calculate such an interval? The endpoints of the interval have to be calculated from the sample, so they are statistics, functions of the sample
圖片參考：http://upload.wikimedia.org/math/c/9/1/c916ca889ccfcef81cf716f93fdcc3d6.png
and hence random variables themselves.
In our case we may determine the endpoints by considering that the sample mean
圖片參考：http://upload.wikimedia.org/math/1/8/e/18e2c3824002002906239140c4560991.png
(grams). By standardizing we get a random variable


圖片參考：http://upload.wikimedia.org/math/4/9/1/4915f6560f256c666a59a254b39e8c68.png

dependent on μ, but with a standard normal distribution independent of the parameter μ to be estimated. Hence it is possible to find numbers −z and z, independent of μ, where Z lies in between with probability 1 − α, a measure of how confident we want to be. We take 1 − α = 0.95. So we have:


圖片參考：http://upload.wikimedia.org/math/e/1/9/e1905b444d6c7d49b5a23df78ef631d2.png

The number z follows from:


圖片參考：http://upload.wikimedia.org/math/9/f/4/9f4cdb8a83c342e2e13eeac43c3a2b0a.png



圖片參考：http://upload.wikimedia.org/math/3/f/0/3f00568339178068af3e30c112a3d697.png

(see probit and cumulative distribution function), and we get:


圖片參考：http://upload.wikimedia.org/math/4/a/6/4a69755f236f9f4fd2bea6542d0e885f.png



圖片參考：http://upload.wikimedia.org/math/0/2/d/02d23babd11a29cdfad846c65063dd19.png


圖片參考：http://upload.wikimedia.org/math/0/c/4/0c4519514a4bc4555e930418785882fa.png





圖片參考：http://upload.wikimedia.org/math/3/3/e/33e5694733f06f8d7ca733621f14338d.png
.
This might be interpreted as: with probability 0.95 one will find the parameter μ between the stochastic endpoints:


圖片參考：http://upload.wikimedia.org/math/b/4/c/b4cfde3aef2551e6ea03938743771e13.png

and


圖片參考：http://upload.wikimedia.org/math/e/1/6/e165b1d2bda9022f6cc54dfee9a60460.png

Every time the measurements are repeated, there will be another value for the mean
圖片參考：http://upload.wikimedia.org/math/9/0/8/9081a3d8bf8f68d6756792ee7eea72c7.png
of the sample. In 95% of the cases μ will be between the endpoints calculated from this mean, but in 5% of the cases it will not be. The actual confidence interval is calculated by entering the measured weights in the formula. Our 0.95 confidence interval becomes:


圖片參考：http://upload.wikimedia.org/math/1/e/9/1e92a8499c3b104c6c702c8d4e2cfa9d.png

This interval has fixed endpoints, where μ might be in between (or not). There is no probability of such an event. We cannot say: "with probability 1 − α the parameter μ lies in the confidence interval." We only know that by repetition in 100(1 − α) % of the cases μ will be in the calculated interval. In 100α % of the cases however it doesn't. And unfortunately we don't know in which of the cases this happens. That's why we say: with confidence level 100(1 − α) % μ lies in the confidence interval."