Usuari:Massimo175/proves

En estadística, l'interval de confiança o error de l'estimació és un interval (un rang entre dos valors) al voltant d'un paràmetre mostral en els quals, amb una probabilitat (o nivell de confiança) determinat, se situarà el aquell paràmetre en la població.

Un paràmetre mostral del qual se sol determinar l'interval de confiança és la mitjana.

El nivell de confiança desitjat és establert per l'investigador (no és determinat per les dades). Més comunament, s'utilitza el nivell de confiança del 95%. No obstant això, es poden utilitzar altres nivells de confiança, per exemple, el 90% i el 99%.

In statistics, a confidence interval (CI) is a type of interval estimate (of a population parameter) that is computed from the observed data. The confidence level is the frequency (i.e., the proportion) of possible confidence intervals that contain the true value of their corresponding parameter. In other words, if confidence intervals are constructed using a given confidence level in an infinite number of independent experiments, the proportion of those intervals that contain the true value of the parameter will match the confidence level.

Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter. However, the interval computed from a particular sample does not necessarily include the true value of the parameter. Since the observed data are random samples from the true population, the confidence interval obtained from the data is also random. If a corresponding hypothesis test is performed, the confidence level is the complement of the level of significance; for example, a 95% confidence interval reflects a significance level of 0.05. If it is hypothesized that a true parameter value is 0 but the 95% confidence interval does not contain 0, then the estimate is significantly different from zero at the 5% significance level.

Si $\alpha$ és l'error aleatori que es vol cometre, la probabilitat serà de 1 - $\alpha$.

A menor nivell de confiança l'interval serà més precís, però es cometrà un major error. Per a comprendre les següents fórmules, és necessari conèixer els conceptes de variabilitat del paràmetre, error, nivell de confiança, valor crític i valor α.

Un interval de confiança és, doncs, una expressió del tipus [θ₁, θ₂] ó θ₁ ≤ θ ≤ θ₂, on θ és el paràmetre a estimar. Aquest interval conté el paràmetre estimat amb una determinada certesa o nivell de confiança 1-α.

Quan s'ofereix un interval de confiança es dóna per descomptat que les dades poblacionals es distribuïxen d'una manera determinada. És habitual que ho facin mitjançant la distribució normal. La construcció d'intervals de confiança també es pot realitzar usant la desigualtat de Txebixev.

Referències

The desired level of confidence is set by the researcher (not determined by data). Most commonly, the 95% confidence level is used. However, other confidence levels can be used, for example, 90% and 99%.

Factors affecting the width of the confidence interval include the size of the sample, the confidence level, and the variability in the sample. A larger sample size normally will lead to a better estimate of the population parameter.

Confidence intervals were introduced to statistics by Jerzy Neyman in a paper published in 1937.

Conceptual basis

In this bar chart, the top ends of the brown bars indicate observed means and the red line segments ("error bars") represent the confidence intervals around them. Although the error bars are shown as symmetric around the means, that is not always the case. It is also important that in most graphs, the error bars do not represent confidence intervals (e.g., they often represent standard errors or standard deviations)

Introduction

Interval estimates can be contrasted with point estimates. A point estimate is a single value given as the estimate of a population parameter that is of interest, for example, the mean of some quantity. An interval estimate specifies instead a range within which the parameter is estimated to lie. Confidence intervals are commonly reported in tables or graphs along with point estimates of the same parameters, to show the reliability of the estimates.

For example, a confidence interval can be used to describe how reliable survey results are. In a poll of election–voting intentions, the result might be that 40% of respondents intend to vote for a certain party. A 99% confidence interval for the proportion in the whole population having the same intention on the survey might be 30% to 50%. From the same data one may calculate a 90% confidence interval, which in this case might be 37% to 43%. A major factor determining the length of a confidence interval is the size of the sample used in the estimation procedure, for example, the number of people taking part in a survey.

Meaning and interpretation

Vegeu també: § Practical Example Interpretation

Various interpretations of a confidence interval can be given (taking the 90% confidence interval as an example in the following).

• The confidence interval can be expressed in terms of samples (or repeated samples): "Were this procedure to be repeated on numerous samples, the fraction of calculated confidence intervals (which would differ for each sample) that encompass the true population parameter would tend toward 90%."
• The confidence interval can be expressed in terms of a single sample: "There is a 90% probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter." Note this is a probability statement about the confidence interval, not the population parameter. This considers the probability associated with a confidence interval from a pre-experiment point of view, in the same context in which arguments for the random allocation of treatments to study items are made. Here the experimenter sets out the way in which they intend to calculate a confidence interval and to know, before they do the actual experiment, that the interval they will end up calculating has a particular chance of covering the true but unknown value. This is very similar to the "repeated sample" interpretation above, except that it avoids relying on considering hypothetical repeats of a sampling procedure that may not be repeatable in any meaningful sense. See Neyman construction.
• The explanation of a confidence interval can amount to something like: "The confidence interval represents values for the population parameter for which the difference between the parameter and the observed estimate is not statistically significant at the 10% level". In fact, this relates to one particular way in which a confidence interval may be constructed.

Categoria:Estadística paramètrica Categoria:Mesura