Confidence Intervals

[A, SfS] Chapter 5: Confidence Intervals: 5.1: Confidence Intervals

Confidence Intervals

Given a set of data, find a range of values that are plausible within a desired level of confidence for the value of a population parameter associated with some variable.

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Let #X_1,...,X_n# denote #n# independent planned measurements of some random variable whose probability distribution involves some parameter #\theta# whose value is unknown to us, and let #\alpha# be some number in the interval #(0,1)#. We want to estimate the value of #\theta# with an interval of the most plausible values for #\theta#, given the eventual data.

Our goal is to find two functions of the measurements #L = L(X_1, \ldots, X_n)# and #U = U(X_1, \ldots, X_n)# such that \[ \mathbb{P}(L \leq \theta \leq U) = 1 - \alpha \] Hence there is a probability of #1 - \alpha# that our eventual measurements will produce an interval #(L,U)# that contains the true value of #\theta#.

This implies that if we were to take a very, very large number of samples and compute #L# and # U# for each sample, approximately #(1 - \alpha)100\%# of the time it would be true that #L \leq \theta \leq U#. Since we don't know the actual value of #\theta#, we wouldn't know which of those intervals #(L,U)# contain #\theta# and which do not, but the laws of probability would assure us about #(1 - \alpha)100\%# of them contain #\theta#.

However, we eventually have only one sample, from which we have the observed data #x_1,...,x_n#. If we use the two functions #L# and #U# to compute the lower boundary #l# and the upper boundary #u# based on these data to obtain the interval #(l,u)#, we cannot be #100\%# certain that #(l,u)# contains #\theta#, but we can say that we have #(1 - \alpha)100\%# confidence that #(l,u)# contains the true value of #\theta#, and call the interval #(l,u)# a #(1 - \alpha)100\%# confidence interval for #\theta#. This gives us a range #(l,u)# of values based on the sample which are plausible values for #\theta#, with a desired confidence level #1 - \alpha#.

Confidence Interval

A confidence interval (CI) for a parameter #\theta# is a range of values, based on sample data, which are highly plausible candidates for the true value of #\theta#.

Confidence intervals are always accompanied by a corresponding confidence level. The confidence level #(1-\alpha)# is the probability that a planned random sample will produce a confidence interval that encloses the target parameter.

From a practical perspective, the confidence level identifies the fraction of the time, in repeated sampling, that the constructed confidence intervals will contain the true value of the parameter.

Suppose you are given a #95\%# CI for some parameter #\theta# that was computed using a specific procedure based on sample data.

This means that, before the sample was selected, there was a #0.95# probability that a sample would be selected that would produce a CI containing the true value of #\theta#.

This implies that if we were to take #100# simple random samples from the population and use the same procedure to compute a #95\%# CI for each of those samples, we would expect about #95# of the CIs to contain the true value of the parameter and about #5# of them not to contain it.

Of course, as long as the true value of the parameter is unknown, we cannot tell which CIs contain the target parameter and which do not.

Interpretation of a Confidence Interval

Confidence intervals are often misinterpreted. Once a confidence interval has been computed from sample data, it is no longer correct to use the word "probability" in connection with the confidence interval.

It is thus incorrect to state: "There is a #95\%# probability that the true value of the parameter is contained within the confidence interval." After all, the true value of the parameter is not random; it is either in the confidence interval or it is not.

What we can say, however, is: "We are #95\%# confident that the true value of the parameter is contained within the confidence interval."

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A confidence interval is typically constructed around a point estimate #\hat{\theta}# for #\theta#, such as the Maximum Likelihood Estimator. The precision of a confidence interval is determined by its margin of error.

Margin of Error

The margin of error of a CI refers to the distance from #\hat{\theta}# to either boundary of the CI, so that oftentimes the CI is expressed instead as #\hat{\theta}# plus or minus its margin of error (e.g., #65\pm 4#).

Manipulating the Margin of Error

Increasing the sample size produces a decrease in the margin of error, and increasing the confidence level produces an increase in the margin of error.

The most common choice for #\alpha# is #0.05#, so that the most commonly-used CI is the #95\%# CI. For greater confidence, the #99\%# CI is often used (#\alpha = 0.01#). However, increasing the confidence level results in a wider CI, and hence less precision. (A #100\%# CI would be #(-\infty,\infty)#, and so completely useless.)

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Confidence Bound

It is also possible to construct a one-sided #(1 - \alpha)100\%# confidence bound for #\theta#.

This could be an upper #(1 - \alpha)100\%# confidence bound for #\theta#, which gives us a confidence interval of the form #(-\infty,u)#.

Or it could be a lower #(1 - \alpha)100\%# confidence bound for #\theta#, which gives us a confidence interval of the form #(l,\infty)#.

We now consider several common settings in which a CI is computed, depending on different research questions.