Confidence Intervals

Interpretation

A confidence interval is a range in which an unknown population parameter is likely to be included. After independent samples are randomly selected from the same population, one confidence interval is constructed based on one sample with a certain confidence level. Together, all the confidence intervals should include the population parameter with the confidence level.

Suppose one is interested in estimating the proportion of bass among all types of fish in a lake. A 95% confidence interval for this proportion, [25%, 36%], is constructed on the basis of a random sample of fish in the lake. After more independent random samples of fish are selected from the lake, through the same procedure more confidence intervals are constructed. Together, all these confidence intervals will contain the true proportion of bass in the lake approximately 95% of the time.

The lower and upper boundaries of a confidence interval are called lower confidence limit and upper confidence limit, respectively. In the earlier example, 25% is the lower 95% confidence limit, and 36% is the upper 95% confidence limit.

Confidence Interval versus Significance Test

A significance test can be achieved by constructing confidence intervals. One can conclude whether a test is significant based on the confidence intervals. Suppose the null hypothesis is that the population mean μ equals 0 and the predetermined significance level is α. Let I be the constructed 100(1 −α)% confidence interval for μ. If 0 is included in the interval I, then the null hypothesis is accepted; otherwise, it is rejected. For example, a researcher wants to test whether the result of an experiment has a mean 0 at 5% significance level. After the 95% confidence interval [− 0.31, 0.21] is obtained, it can be concluded that the null hypothesis is accepted, because 0 is included in the confidence interval. If the 95% confidence interval is [0.11, 0.61], it indicates that the mean is significantly different from 0.

For a predetermined significance level or confidence level, the ways of constructing confidence intervals are usually not unique. Shorter confidence intervals are usually better because they indicate greater power in the sense of significance test.

One-sided significance tests can be achieved by constructing one-sided confidence intervals. Suppose a researcher is interested in an alternative hypothesis that the population mean is larger than [Page 212]0 at the significance level .025. The researcher will construct a one-sided confidence interval taking the form of (−∞, b] with some constant b. Note that the width of a one-sided confidence interval is infinity. Following the above example, the null hypothesis would be that the mean is less than or equal to 0 at the .025 level. If the 97.5% one-sided confidence interval is (−∞,3.8], then the null hypothesis is accepted because 0 is included in the interval. If the 97.5% confidence interval is (−∞, −2.1] instead, then the null hypothesis is rejected because there is no overlap between (−∞, −2.1] and [0, ∞).

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