Topic 16 Interval Estimation - University of Arizona

Overview

Means

Topic 16 Interval Estimation

Confidence Intervals for Means

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Overview

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Outline

Overview

Means z intervals t intervals Two Sample t intervals

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Overview

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Overview

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 by a statistic that is subset C^ (x) of the parameter space. We consider both the classical and Bayesian approaches.

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Overview

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Overview

The random set C^ (X ) is chosen to have a prescribed high probability, , of containing

the true parameter value .

P{ C^ (X )} = .

C^ (x) is called a -level confidence set. For a single parameter, the typical choice of

confidence set is a confidence interval

C^ (x) = (^ (x), ^u(x)).

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.

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For X1.X2. . . . Xn normal random variables, unknown mean ?, known variance 02,

X? - ? Z=

0/ n

is a standard normal. For any between 0 and 1, let z satisfy

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P{Z > z} = 0.35

or equivalently

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0.25

P{Z z} = 1 - .

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0.15

The value is known as the upper tail prob- 0.1

ability with critical value z. We compute 0.05

area

this in R with qnorm(0.975) for = 0.025.

0

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-2

-1

0

1

2

z

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