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
2
• 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
u
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• most of these intervals do contain the true mean but two do not
• there is an important connection between confidence intervals and hypothesis
3
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