STATISTICS 321 – Dr. Soma Roy
Chapter 5: Confidence Intervals
About time we looked at some serious applications of the stuff we have been studying all quarter. Consider
the following questions.
1. What % of Americans approve of the way President Obama is handling his job?
2. What % of M and M’s are orange?
3. What is the average speed of vehicles traveling on Highway 101?
How do we answer these and other questions? That is, if we only have the opportunity to make one guess
as to what the parameter equals, we would give a point estimate. Could we, however, use the knowledge
about the standard error of estimators, to get an estimate of a range of values that might be plausible
for the parameter?
We could use a
confidence interval
.
A confidence interval is an interval of
The general form of a confidence interval for a parameter is given by:
statistic
±
margin of error
Confidence intervals for the population mean,
μ
Let us look at the following example:
Let
X
1
, X
2
, . . . , X
n
be a random sample from Normal(
μ, σ
2
)
,
where
μ
is unknown and
σ
2
is known.
Now, this is a pretty big stretch in terms of the validity of these assumptions.
Especially, since if we
know something about the variance we usually know something about the mean.
For the time being,
however, let’s just go with this. Later on we will learn methods of inference that do not require these
semiunrealistic assumptions.
Our goal is to make inferences about the population mean,
μ
. Our intuitive estimator (which happens to
be an unbiased MLE) is
, whose distribution can be given by :
This is called the
of
¯
X
.
Recall the Empirical Rule. It says,
P
(
μ

2
σ
√
n
<
¯
X < μ
+ 2
σ
√
n
)
≈
0
.
95
Also, we can verify that
P
(
μ

2
σ
√
n
<
¯
X < μ
+ 2
σ
√
n
)
=
P
(

2
<
¯
X

μ
σ
√
n
<
2)
=
P
(

2
< Z <
2)
≈
0
.
95
1
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This means that for about 95% of the samples, were going to get a sample average,
¯
X
, that is within two
standard deviations of the true population mean. Depending on how big our standard deviation is, thats
not bad. Right?
Looking at the above from
μ
’s point of view:
P
(
μ

2
σ
√
n
<
¯
X < μ
+ 2
σ
√
n
)
≈
0
.
95
Then,
P
(
μ

2
σ
√
n
<
¯
X < μ
+ 2
σ
√
n
)
=
P
(

2
σ
√
n
<
¯
X

μ <
2
σ
√
n
)
=
P
(

¯
X

2
σ
√
n
<

μ <

¯
X
+ 2
σ
√
n
)
=
P
(
¯
X

2
σ
√
n
< μ <
¯
X
+ 2
σ
√
n
)
≈
0
.
95
New Interpretation:
A 95
%
Confidence Interval for
μ
:
is given by
Example:
Fruit bats
The Giant goldencrowned flying fox (Acerodon jubatus), also known as the
Goldencapped fruit bat, is a rare fruit bat said to be the largest bat in the world. (For more info check
Wikipedia.) Assume that the distribution of wingspans for the flying fox is normally distributed with
standard deviation 7 inches. The average wingspan of a random sample of 20 bats is 58 inches. Compute
a 95% confidence interval for the average wingspan of all goldencrowned flying foxes.
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 Winter '08
 Hascall
 Normal Distribution, Standard Deviation

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