
In statistics, a confidence interval (CI) is a particular kind of interval estimate of a population parameter. Instead of estimating the parameter by a single value, an interval likely to include the parameter is given. Thus, confidence intervals are used to indicate the reliability of an estimate. How likely the interval is to contain the parameter is determined by the confidence level or confidence coefficient. Increasing the desired confidence level will widen the confidence interval.
A confidence interval is always qualified by a particular confidence level, usually expressed as a percentage; thus one speaks of a "95% confidence interval". The end points of the confidence interval are referred to as confidence limits.

The gambler's fallacy : If a fair coin is tossed repeatedly and tails comes up a larger number of times than is expected, a gambler may incorrectly believe that this means that heads is more likely in future tosses
The reversal is also a fallacy, the reverse gambler's fallacy, in which a gambler may instead decide that tails are more likely out of some mystical preconception that fate has thus far allowed for consistent results of the tail; the false conclusion being, why change if odds favor tails? Again, the fallacy is the belief that the "universe" somehow carries a memory of past results which tend to favor or disfavor future outcomes.

If the central limit theorem didn’t exist, it would not be possible to use statistics. We would be unable to reliably estimate a parameter like the mean by using an average derived from a much smaller sample. This would all but shut down research in the social sciences and the evaluation of new drugs since these depend on statistics. It would invalidate the use of polls and completely alter the nature of marketing research not to mention politics.
Thanks to the central limit theorem, we can be sure that a mean or xbar based on a reasonably large randomly chosen sample will be remarkably close to the true mean of the population. If we need more certainty we need only increase the sample size. What’s more, it does not matter if we are characterizing a city, state, or the entire United States, we can use the same sample size. It will give the same level of certainty regardless of the population size.
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