Chapter 6: Introduction to Inference
This chapter concerns inference procedures for
the population average. You will learn about
confidence intervals
and
significance tests to
learn about a population average
μ
. The world
uses both of these methods and you need to
know both also.
6.1
Estimating with Confidence
If the entire population of SAT scores has mean
μ
and standard deviation
σ
, then in repeated
samples of size 500 the sample mean
x
has
a N(
μ
,
500
σ
) distribution. Let us suppose
that we know that the standard deviation
σ
of
SATM scores in California population is
100
=
σ
.
In repeated sampling the sample
mean
x
follows the normal distribution
centered at the unknown population mean
μ
and having standard deviation
5
.
4
500
100
=
=
x
σ
The 68-95-99.7 rule says that the probability is
about 0.95 that
x
will be within 9 points (
σ
×
2
)
of the population mean score
μ
.
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We say that we are 95% confident that the
unknown mean score for a California seniors lies
between
452
9
461
9
=
-
=
-
x
and
470
9
461
9
=
+
=
+
x
.
Figure
6.2
x
=461
lies within
9
±
of
μ
in 95%
of
all samples, so
μ
also lies within
9
±
of
x
in
samples from more than 250,000 high school
seniors in California.
Confidence
intervals
We will use
C
to stand for the confidence level in
decimal form. For example, a 95% confidence
level corresponds to
C
=0.95.
Confidence Interval
A level
C
confidence interval
for a parameter
is an interval computed from sample data by a
method that has probability
C
of producing an
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- Spring '09
- Statistics, Normal Distribution, Tim Kelly
-
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