# Relationship between confidence interval and precision

### Estimation and Confidence Intervals The confidence level is the probability that the interval actually is a close relationship between confidence and precision, these terms are not. Difference between Confidence Level and Confidence Interval methods like correlation, regression etc, to arrive at intervals for the required confidence level. The precision refers to the width of a confidence interval. Tomorrow's temperature is probably going to be somewhere between F and F, however is it.

The parameter is an unknown constant, and no probability statement concerning its value may be made Seidenfeld's remark seems rooted in a not uncommon desire for Neyman-Pearson confidence intervals to provide something which they cannot legitimately provide; namely, a measure of the degree of probability, belief, or support that an unknown parameter value lies in a specific interval. Following Savagethe probability that a parameter lies in a specific interval may be referred to as a measure of final precision.

### Accuracy vs. Precision of confidence intervals

While a measure of final precision may seem desirable, and while confidence levels are often wrongly interpreted as providing such a measure, no such interpretation is warranted. Admittedly, such a misinterpretation is encouraged by the word 'confidence'. A confidence interval is not a definitive range of plausible values for the sample parameter, though it may be understood as an estimate of plausible values for the population parameter.

Philosophical issues[ edit ] The principle behind confidence intervals was formulated to provide an answer to the question raised in statistical inference of how to deal with the uncertainty inherent in results derived from data that are themselves only a randomly selected subset of a population. There are other answers, notably that provided by Bayesian inference in the form of credible intervals. Confidence intervals correspond to a chosen rule for determining the confidence bounds, where this rule is essentially determined before any data are obtained, or before an experiment is done.

The rule is defined such that over all possible datasets that might be obtained, there is a high probability "high" is specifically quantified that the interval determined by the rule will include the true value of the quantity under consideration.

The Bayesian approach appears to offer intervals that can, subject to acceptance of an interpretation of "probability" as Bayesian probabilitybe interpreted as meaning that the specific interval calculated from a given dataset has a particular probability of including the true value, conditional on the data and other information available. The confidence interval approach does not allow this since in this formulation and at this same stage, both the bounds of the interval and the true values are fixed values, and there is no randomness involved.

On the other hand, the Bayesian approach is only as valid as the prior probability used in the computation, whereas the confidence interval does not depend on assumptions about the prior probability. The questions concerning how an interval expressing uncertainty in an estimate might be formulated, and of how such intervals might be interpreted, are not strictly mathematical problems and are philosophically problematic. In the physical sciencesa much higher level may be used.

If two confidence intervals overlap, the two means still may be significantly different. Such an approach may not always be available since it presupposes the practical availability of an appropriate significance test. Naturally, any assumptions required for the significance test would carry over to the confidence intervals.

It may be convenient to make the general correspondence that parameter values within a confidence interval are equivalent to those values that would not be rejected by a hypothesis test, but this would be dangerous.

In many instances the confidence intervals that are quoted are only approximately valid, perhaps derived from "plus or minus twice the standard error," and the implications of this for the supposedly corresponding hypothesis tests are usually unknown. It is worth noting that the confidence interval for a parameter is not the same as the acceptance region of a test for this parameter, as is sometimes thought.

The confidence interval is part of the parameter space, whereas the acceptance region is part of the sample space.

For the same reason, the confidence level is not the same as the complementary probability of the level of significance. You can think about the red interval that is plotted on this figure and imagine that it extends even further. It would be much likely for it to then capture the true population parameter which is shown here as the vertical dashed line.

Therefore, as the confidence level increase, so does the width of the confidence interval. Therefore, as we increase the confidence level, the width of the interval increases as well. More accurate means a higher confidence level.

Confidence intervals and margin of error - AP Statistics - Khan Academy

So if we are saying that we want to increase accuracy, we also need to increase the confidence level, but this might come at a cost. What is the drawback when using a wider interval As the confidence level increase, the width of the confidence interval increase as well.

Which then increase the accuracy. However, the precision goes down. Suppose you are watching the weather forecase, and you are told that the next day, low is F and high is F. Tomorrow's temperature is probably going to be somewhere between F and F, however is it informative?

## Estimation and Confidence Intervals

Or, in other wards, is it precise? It is nearly impossible to figure out what to wear tomorrow according to this information. Inthe survey collected responses from 1, US residents. Determine if each of the following statements are true or false.