From a data set with
n
observations calculate the sample mean
x
and sample standard deviation
s
.
A
)
1
(
α

100
% confidence interval estimate for the population mean
is given by:
n
s
t
x
c
±
where
c
t
is the critical value from the tdistribution with (
n

1
)
degrees of freedom such that:
2
)
t
t
(
P
c
)
1
n
(
α
=
>

Econ 325 – Chapter 7
26
Example:
Gasoline Consumption of Trucks
A data set has observations on fuel consumption, in miles per gallon,
for 24 trucks.
Summary statistics are:
18.68
=
x
and
1.695
=
s
A 90% confidence interval estimate for the population mean fuel
consumption is:
24
1.695
18.68
c
t
±
The graph below illustrates the tdistribution critical value.
PDF of
)
23
(
t
tc
0
tc
Area = 0.9
Upper Tail Area = 0.05
Lower Tail Area = 0.05
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Econ 325 – Chapter 7
27
The Appendix Table for the tdistribution can be used to lookup the
critical value
c
t
.
For this example, to correctly use the table, select
the degrees of freedom
(
n

1
) = 23, and set the upper tail area to
0.10/2 = 0.05.
Alternatively, use Microsoft Excel.
Select Insert Function:
TINV(0.10, 23)
This returns the answer:
1.714
=
c
t
The calculations required for the interval estimate are:
24
1.695
1.714
18.68
⋅
±
For the given data set, the calculations give a 90% confidence interval
estimate for the population mean as:
[
19.27
18.09
,
]
Econ 325 – Chapter 7
28
Chapter 7.7 Sample Size Determination
A wide confidence interval reflects uncertainty about the parameter
being estimated.
A larger sample size
n
will give a narrower
interval.
Consider a confidence interval for the population mean
μ
in a
situation where the population variance
2
σ
is known from previous
research.
For a given data set, a
)
1
(
α

100
%
interval estimate for the
population mean is:
[
n
z
x
,
n
z
x
c
c
σ
+
σ

]
where
c
z
is the value such that:
2
1
)
z
(
F
)
z
Z
(
P
c
c
α

=
=
<
The width of the interval estimate is:
n
z
2
w
c
σ
⋅
=
Suppose a set width
w
is desired.
What sample size
n
will guarantee this width ?
Rearranging gives:
w
z
2
n
c
σ
⋅
=
By squaring both sides:
(
)
2
c
w
z
2
n
σ
⋅
=
Round up to get an integer number for
n
.
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 Fall '10
 WHISTLER
 Normal Distribution, Standard Deviation, Variance, interval estimate, Point Estimators

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