Confidence Interval

A 100 (1 − α)% confidence interval is an interval estimate around a POPULATION parameter θ that, under repeated random samples of size N, would be expected to include θ's true value 100 (1 − α)% of the time. The confidence interval is a natural adjunct to PARAMETER ESTIMATION because it indicates the precision with which a POPULATION PARAMETER is estimated by a sample statistic.

The confidence level, 100 (1 − α)%, is chosen a priori. A TWO-SIDED confidence interval uses a lower limit L and upper limit U that each contain θ's true value 100 (1 − α/2)% of the time, so that together they contain θ's true value 100 (1 − α)% of the time. This interval often is written as [L,U], and sometimes writers combine a confidence level and interval by writing Pr(L<θ<U) = 1 − α. In some applications, a one sided confidence interval is used. Confidence intervals may be computed for a large variety of population parameters such as the MEAN, VARIANCE, proportion, and R-SQUARED.

The confidence interval is closely related to the SIGNIFICANCE TEST because a 100 (1 − α)% confidence interval includes all hypothetical values of the population parameter that cannot be rejected by a significance test using a significance criterion of α. In this respect, it provides more information than a significance test does. Confidence intervals become narrower with larger sample size and/or lower confidence levels.

The limits L and U are derived from a sample statistic (often the sample estimate of θ) and a sampling distribution that specifies the probability of getting each value that the sample statistic can take. Thus, L and U also are sample statistics and will vary from one sample to another. Suppose, for example, that a standard IQ test has been administered to a random [Page 169]sample of N = 15 adults from a large population with a sample mean of 104 and standard deviation (SD) of 18. We will construct a two-sided 95% confidence interval for the mean, μ. The limits U and L must have the property that, given a significance criterion of α, sample size of 15, mean of 104, and standard deviation of 18, we could reject the hypotheses that μ> 104 + U or μ< 104 − L but not L<μ<U. Figure 1 displays those limits.

The sampling distribution of the T ratio, where serr stands for the standard error of the mean, defined by

is a t-distribution with df = N − 1 = 14. When df = 14, the value tα/2 = 2.145 standard error units above the mean cuts α/2 = .025 from the upper tail of this t-distribution; likewise, −tα/2 =−2.145 standard error units below the mean cuts α/2 = .025 from the lower tail. The sample standard error is serr = SD/√N = 4.648. So a t-distribution around 104 + (2.145)(4.648) = 113.97 has .025 of its tail below 104, while a t-distribution around L = 104−(2.145)(4.648) = 94.03 has .025 of its tail above 104. These limits fulfill the above required property, so the 95% confidence interval for μ is [94.03, 113.97].

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