8.2 A Confidence Interval For The Population Mean μ (σ Known)
In attempting to estimate μ from a simple random sample of size n, we previously employed the sample mean as a point estimator for μ, where a point estimator reports a single numerical value as the estimate of μ. Now, let us report a whole range of possible values rather than a single point estimate. This range of values is called an interval estimate or confidence interval for μ, that is, it is a range of values that enables us to state just how confident we are that the reported interval contains μ. What is the role of a confidence interval for μ? It indicates how precisely μ has been estimated from the sample; the narrower the interval, the more precise the estimate.
As we shall now see, we can view a confidence interval for μ as a “generalization of the error bound concept.” In this regard, we shall eventually determine our confidence limits surrounding μ as
where now the term ± error bound is taken to be our degree of precision.
Let us see how all this works. To construct a confidence interval for μ, we need to find two quantities L1 and L2 (both function of the sample values) such that, before the random sample is drawn,
where and are, respectively, lower and upper confidence limits for μ and is the ...