Topic 16 Interval Estimation - University of Arizona

Overview

Means

Correspondence between Two-Sided Tests and Confidence Intervals

Interpretation of the Confidence Interval

Topic 16 Interval Estimation

Confidence Intervals for Means

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Overview

Means

Correspondence between Two-Sided Tests and Confidence Intervals

Interpretation of the Confidence Interval

Outline

Overview

Means z intervals t intervals

Correspondence between Two-Sided Tests and Confidence Intervals Two Sample t intervals

Interpretation of the Confidence Interval

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Overview

Means

Correspondence between Two-Sided Tests and Confidence Intervals

Overview

Interpretation of the Confidence Interval

The quality of an estimator can be evaluated using its bias and variance. Often, knowledge of the distribution of the estimator and this allows us to take a more comprehensive statement about the estimation procedure.

For interval estimation, given data x, we replace the point estimate ^(x) for the parameter , the parameter space by a statistic that is subset C^(x) . We consider both the classical and Bayesian approaches.

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Overview

Means

Correspondence between Two-Sided Tests and Confidence Intervals

Interpretation of the Confidence Interval

Overview

For a given parameter value , the coverage probability of C^(X ) is

P{ C^(X )}, The C^(X ) is typically chosen to have a prescribed high probability, , of containing the true parameter value .

P{ C^(X )} for all , C^(x) is called a -level confidence set. For a single parameter, the typical choice of confidence set is a confidence interval. This can be two-sided.

C^(x) = {; ^ (x), ^u(x)} = [^ (x), ^u(x)].

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Overview

Means

Correspondence between Two-Sided Tests and Confidence Intervals

Interpretation of the Confidence Interval

Overview

Often this interval takes the form [^(x) - m(x), ^(x) + m(x)] = ^(x) ? m(x) where the two statistics,

? ^(x) is a point estimate, and m(x) is the margin of error.

For one-sided confidence intervals, we can have

C^(x) = {; ^u(x)} = (-, ^u(x)]. where ^u(x) is called the upper confidence bound or

C^(x) = {; ^u(x) } = [^ (x), ). where ^ (x) is called the lower confidence bound

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