Relation To Hypothesis Testing
While the formulations of the notions of confidence intervals and of statistical hypothesis testing are distinct they are in some senses related and to some extent complementary. While not all confidence intervals are constructed in this way, one general purpose approach to constructing confidence intervals is to define a 100(1 − α)% confidence interval to consist of all those values θ0 for which a test of the hypothesis θ = θ0 is not rejected at a significance level of 100α%. Such an approach may not always be available since it presupposes the practical availability of an appropriate significance test. Naturally, any assumptions required for the significance test would carry over to the confidence intervals.
It may be convenient to make the general correspondence that parameter values within a confidence interval are equivalent to those values that would not be rejected by a hypothesis test, but this would be dangerous. In many instances the confidence intervals that are quoted are only approximately valid, perhaps derived from "plus or minus twice the standard error", and the implications of this for the supposedly corresponding hypothesis tests are usually unknown.
It is worth noting, that the confidence interval for a parameter is not the same as the acceptance region of a test for this parameter, as is sometimes thought. How could it be? The confidence interval is part of the parameter space, whereas the acceptance region is part of the sample space. For the same reason the confidence level is not the same as the complementary probability of the level of significance.
Read more about this topic: Confidence Interval
Famous quotes containing the words relation to, relation, hypothesis and/or testing:
“In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
—Simone Weil (1909–1943)
“The problem of the twentieth century is the problem of the color-line—the relation of the darker to the lighter races of men in Asia and Africa, in America and the islands of the sea. It was a phase of this problem that caused the Civil War.”
—W.E.B. (William Edward Burghardt)
“It is more than likely that the brain itself is, in origin and development, only a sort of great clot of genital fluid held in suspense or reserved.... This hypothesis ... would explain the enormous content of the brain as a maker or presenter of images.”
—Ezra Pound (1885–1972)
“Traditional scientific method has always been at the very best 20-20 hindsight. It’s good for seeing where you’ve been. It’s good for testing the truth of what you think you know, but it can’t tell you where you ought to go.”
—Robert M. Pirsig (b. 1928)