Confidence Intervals  I
• The last topic in our “Review of Statistics”
is
Confidence Intervals
.
• Recall that we have an estimate,
Y
, of
μ
Y
.
Again,
Y
will likely be fairly close to
μ
Y,
,
but it will generally not equal
μ
Y
.
Given
our estimate
Y
, we might want to try to
quantify the range that
μ
Y
might be in.
• We can do this using Confidence Intervals.
Confidence Intervals  II
• Given our estimate
Y
, a 95% confidence interval for
μ
Y
is given by:
• Statistically, this interval will contain (or “cover”)
the true population mean
μ
Y
95% of the time.
Hence, we can say that we are 95% confident that
μ
Y
is in this interval or region.
• Confidence intervals are a very useful and intuitive
way of describing what we have learned
statistically.
22
1.96
,
1.96
YY
ss
nn
⎛⎞
−+
⎜⎟
⎝⎠
Confidence Intervals  III
• Example:
Suppose
Y=
84,
s
2
Y
is 25, and
n
= 100.
• A 95% confidence interval for
μ
Y
is:
• We would conclude that we are 95% sure that
μ
Y
is
in this range.
()
100
100
84 1.96
, 84 1.96
25
25
80.08 , 87.92
=
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View Full DocumentConfidence Intervals  IV
•N
o
t
e
s
:
•
1) Confidence interval shrinks (i.e. one gets a more
precise interval) as:
–
n
increases (all else equal)
–
s
2
Y
decreases (all else equal)
•
2) Can also do 90% or 99% confidence intervals:
–9
0
%
C
I
:
9
%
C
I
:
•
To be more confident that your interval covers
μ
Y
, you
need a bigger interval!
22
1.64
,
1.64
YY
ss
nn
⎛⎞
−+
⎜⎟
⎝⎠
2.58
2.58
Linear Regression  I
• We now move to studying the “Linear Regression
Model with One Regressor”.
Regression models
study the relationship
between
variables.
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 Spring '07
 SandraBlack
 Statistics, Econometrics, Regression Analysis, Standardized test, Yi

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