Topic 16 Interval Estimation - University of Arizona
Overview
Means
Topic 16 Interval Estimation
Confidence Intervals for Means
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Overview
Means
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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Overview
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Means
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
0.4
P{Z > z} = 0.35
or equivalently
0.3
0.25
P{Z z} = 1 - .
0.2
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
-3
-2
-1
0
1
2
z
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