Interval Estimate

Interval estimates aim at estimating a parameter using a range of values rather than a single number. For example, the proportion of people who voted for a particular candidate is estimated to be 43% with a margin of error of three (3.0) percentage points based on a political poll. From this information, an interval estimate for the true proportion of voters who favored the candidate would then consist of all the values ranging from a low of 40% to a high of 46%—which is usually presented as (0.40, 0.46). If the interval estimate is derived using the probability distribution of the point estimate, then the interval estimate is often referred to as a “confidence interval” where the “confidence coefficient” quantifies the probability that the process and subsequent derivation will produce an interval estimate that correctly contains the true value of the parameter.

While point estimates use information contained in a sample to compute a single numeric quantity to estimate a population parameter, they do not incorporate the variation in the population. Interval estimates, on the other hand, make use of the point estimate along with estimates of the variability in the population to derive a range of plausible values for the population parameter. The width of such intervals is often a function of the “margin of error,” which is itself a function of the degree of confidence, the overall sample size, and sampling design as well as the variability within the population. In practice, intervals that are narrower usually provide more specific and useful information about the location of the population parameter as compared to wider intervals that are often less informative or more generic (e.g. the population proportion of voters in favor of a candidate is between 0 and 1 would be an interval estimate that is not informative).

Interval estimates can be derived for any population parameter, including proportions, means, totals, quan-tiles, variances, regression parameters, and so on. The generic format of an interval estimate for the population parameter θ can be written as where DV represents a distribution value determined [Page 362]by the sampling distribution of the estimator 9, and SE refers to the standard error of the estimator. Many interval estimates are in fact symmetric around the corresponding point estimate (i.e. as is generally true for means, totals, and proportions based on simple or stratified random sampling designs), but this property is not universal. For example, if interest is given to estimating the variability in birth weights in a population using a simple random sample of hospital birth records, then estimates of σ2 will be based on a chi-squared distribution and thus will generally be asymmetric about the point estimate.

As example, suppose that interest is given in estimating the average household 6-month out-of-pocket dental expenses within a midwestern U.S. state. An interval estimate is to be derived using a sample of 10 households that are randomly selected from each of eight geographically denned strata. Sample means, variances, and stratum sizes (i.e. numbers of households) are provided in Table 1.

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