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Confidence Intervals and Hypothesis Tests

Overview: Recall that a point estimator for a parameter of interest is a statistic which is a function of the random variables from which our sampling method chooses values. The properties of such an estimator, such as having a small bias or a small mean square error, are usually dependent only upon certain features of the sampling distribution of the statistic, such as its mean or variance. However, knowing (or assuming) the nature of that sampling distribution will allow us to make probability statements about the accuracy of our point estimator or test hypotheses about the value of that parameter. These statements are often be described in terms of “confidence intervals”. Here’s the basic idea. Suppose we have a point estimator Θ̂ = H(X1, X2, . . . , Xn) for a parameter θ of the common distribution of the random sample X1, X2, . . . , Xn. Using the sampling distribution of this point estimator assuming a particular value of the parameter, we can construct a statistic with values which are the endpoints of an interval which will contain that value of the parameter (we often say “cover the parameter”) with a certain probability or confidence level that we can prescribe in advance. After we perform the experiment, we have a numerical interval which can be regarded as an interval estimate of the parameter with our specified confidence level. Alternatively, we can see whether or not this interval covers a hypothesized value for the parameter and decide on this basis whether or not to reject that hypothesis. Let’s give an example of this using one of our most important estimators, the sample mean. Recall that the sample mean