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Chapter 8
CONFIDENCE INTERVALS
8.1 Introduction: The onesample problem for means. The Z interval.
8.2 Conﬁdence interval for a proportion.
8.3 Comparing means of two independent samples.
8.4 Comparing two proportions.
For lecture 13:
8.5 The onesample problem for means with unknown variance and small samples.
Student’s
t
distribution, the Tinterval.
8.6 Two independent samples, small sample sizes, variances unknown.
The twosample
t
statistic.
8.7 Paired samples.
8.8 Conﬁdence intervals for variances. The chisquared distribution.
8.9 Comparing two variances. The
F
distribution.
1
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View Full Document8.1
INTRODUCTION: THE ONESAMPLE PROBLEM FOR MEANS
Ref: Devore 6e, pages 281–289. Consider a random sample
X
1
,X
2
,...,X
n
.
The
{
X
i
}
are independent from a distribution
F
which depends on an unknown parameter,
θ
say. We wish to give an interval estimate for
θ
based on the value of some statistic or
point estimate
ˆ
θ
for
θ
. For example, if
θ
is a mean, we might use
ˆ
θ
=
X
. If
θ
is a variance
we might consider using the sample variance
ˆ
θ
=
S
2
. The conﬁdence interval for
θ
is then
based on the sampling distribution of
ˆ
θ
.
Example: Onesample normal mean problem.
Suppose
X
1
,X
2
,...,X
n
are a random sample from
N
(
μ,σ
2
) where
μ
is unknown and
σ
is
known. We want to get a conﬁdence interval for
μ
. We use
X
and note that the sampling
distribution of
X
is
N
(
μ,
σ
2
n
). Hence by the 68, 95, 99.7% rule:
P
[

2
<
X

μ
σ/
√
n
<
2]
’
0
.
95
(Actually, 2 should be replaced by 1.96 if we want to be exact).
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 Spring '05
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