N

and say that the probability is approximately 0.95, or 95 percent, that intervals like it will include the true \x, we are in fact constructing an interval estimator for fi. Note that the interval given previously is random since it is based on X, which will vary from sample to sample.

More generally, in interval estimation we construct two estimators Q1 and 02, both functions of the sample X values, such that

That is, we can state that the probability is 1 — a that the interval from §1 to Q2 contains the true Q. This interval is known as a confidence interval of size 1 — a for Q, 1 — a being known as the confidence coefficient. If a = 0.05, then 1 — a = 0.95, meaning that if we construct a confidence interval with a confidence coefficient of 0.95, then in repeated such constructions resulting from repeated sampling we shall be right in 95 out of 100 cases if we maintain that the interval contains the true Q. When the confidence coefficient is 0.95, we often say that we have a 95% confidence interval. In general, if the confidence coefficient is 1 — a, we say that we have a 100(1 — a)% confidence interval. Note that a is known as the level of significance, or the probability of committing a Type I error. This topic is discussed in Section A.8.

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