Lecture 26
Nancy Pfenning Stats 1000
Chapter 12: More About Confidence Intervals
Recall: Setting up a confidence interval is one way to perform statistical inference: we use a
statistic
measured
from the
sample
to construct an interval estimate for the unknown
parameter
for the
population
. We learned
in Chapter 10 how to construct a confidence interval for unknown population proportion
p
based on sample
proportion ˆ
p
, when there was a single categorical variable of interest, such as smoking or not.
In this chapter, we will learn how to construct other confidence intervals:
•
for population mean
μ
based on sample mean ¯
x
when there is one quantitative variable of interest;
•
for population mean di
ff
erence
μ
d
based on sample mean di
ff
erence
¯
d
in a matched pairs study when
the single set of (quantitative) di
ff
erences
d
is the variable of interest;
•
for di
ff
erence between population means
μ
1

μ
2
based on di
ff
erence between sample means ¯
x
1

¯
x
2
in
a twosample study.
The latter two situations involve one quantitative variable and an additional categorical variable with
two possible values, although we may think of the distribution of di
ff
erences in the matchedpairs study as
a single quantitative variable.
Also discussed in the textbook but not in our course is the method of constructing a confidence interval for
the di
ff
erence between two population proportions
p
1

p
2
based on the di
ff
erence between sample proportions
ˆ
p
1

ˆ
p
2
. Because such situations involve two categorical variables, they can be handled instead with a chi
square procedure, which will be discussed further in Chapter 15.
The Empirical Rule for normal distributions allowed us to state that in general, the probability is 95%
that a normal variable falls withing 2 standard deviations of its mean. Since sample proportion ˆ
p
for a large
enough sample size
n
is approximately normal with mean
p
and standard deviation
p
(1

p
)
n
, we were able
to construct an approximate 95% confidence interval for
p
: ˆ
p
±
2
ˆ
p
(1

ˆ
p
)
n
.
In general, an approximate 95% confidence interval for a parameter is the accompanying statistic plus or
minus two standard errors; this works well if the statistic’s sampling distribution is approximately normal. If
we are interested in the unknown population mean
μ
when there is a single quantitative variable of interest,
we use the fact (established in Chapter 9) that sample mean ¯
x
has mean
μ
and standard deviation
σ
√
n
.
For a large enough sample size
n
(say,
n
at least 30), population standard deviation
σ
will be fairly well
approximated by sample standard deviation
s
and so our standard error for ¯
x
is
s.e.
(¯
x
) =
s
√
n
. Also for large
n
, by virtue of the Central Limit Theorem, the distribution of ¯
x
will be approximately normal, even if the
underlying population variable
X
is not. Thus, for a large sample size
n
, the Empirical Rule tells us that an
approximate 95% confidence interval for population mean
μ
is
¯
x
±
2
s
√
n
Example
The mean number of credits taken by a sample of 81 statistics students was 15.60 and the standard
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 Fall '06
 taeyoungpark
 Statistics, Normal Distribution, Standard Deviation, Statistical hypothesis testing

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