CS计算机代考程序代写 1

1

Confidence Intervals

• A confidence interval provides a simple summary of how precisely a parameter,
denoted θ, is estimated.

• In many situations, a (1− α)100% confidence interval is of the form

(θ̂ − s
θ̂
tα/2, θ̂ + sθ̂tα/2)

where

– θ̂ is an estimate of θ

– s
θ̂

is its standard error

– tα/2 is the upper α/2th quantile from a distribution like the normal or t

• s
θ̂

is usually inversely proportional to the square root of the sample size, so the
interval is narrower for larger samples

• tα/2 is larger for smaller α or larger confidence level, so a 99% confidence interval
is wider than a 95% confidence interval

• the interval is constructed so that in advance there is probability 1 − α that it
includes the true value of the parameter

• once we get the data and evaluate the interval endpoints we don’t know whether
or not the interval contains the true parameter

– but we are confident that it does

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• the figure below shows 95% confidence intervals for the mean constructed using
25 different random samples

25 Confidence intervals
from normal samples of size 20

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• most of these intervals do contain the true mean but two do not

• there is an important connection between confidence intervals and hypothesis

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tests

– a (1−α)100% confidence interval contains all values θ0 which are not rejected
in a test of H0 : θ = θ0 versus Ha : θ 6= θ0 at level α