3.
The Director of Student Loans wants to know the mean amount owed on
student loans at the time of his/her graduation. The director selects a
sample of 20 graduating students and contacts each to find the information.
298
Characteristics of the
t
distribution
1. It is, like the
z
distribution, a
continuous distribution
.
2. It is, like the
z
distribution,
bellshaped
and
symmetrical
.
3. There is
not one
t
distribution
, but rather
a family of
t
distributions
. All
t
distributions have a mean of 0, but
their standard deviations differ according to the
sample
size,
n.
4. The
t
distribution is more spread out
and
flatter at the
center than the standard normal distribution
As the
sample size increases, however, the
t
distribution
approaches the standard normal distribution
299
Comparing the z and
t
Distributions when
n
is small, 95% Confidence Level
300
Confidence Interval Estimates for the Mean
Use
Z
distribution
If the population standard
deviation is
known OR
the sample is
greater
than or equal to 30
.
Use
t
distribution
If the population standard
deviation is
unknown
AND
the sample is
less than 30
.
300
n
z
X
V
r
n
s
t
X
r
When to Use the
z
or
t
Distribution for
Confidence Interval Computation
301
OR
n
>=
30
Confidence Interval for the Mean –
Example using the
t
distribution
A tire manufacturer wishes to
investigate the tread life of its tires.
A
sample of 10
tires driven 50,000
miles revealed a
sample mean of
0.32 inch
of tread remaining with a
standard deviation of 0.09
inch.
Construct a 95 percent confidence
interval for the population mean.
Would it be reasonable for the
manufacturer to conclude that after
50,000 miles the population mean
amount of tread remaining is 0.30
inches?
n
s
t
X
s
x
n
n
1
,
2
/
unknown)
is
(since
dist.

t
the
using
C.I.
the
Compute
09
.
0
32
.
0
10
:
problem
in the
Given
0
r
!
!
!
D
V
301
NOTE:
Degrees of Freedom (df) = n  1
Student’s
t
distribution Table
302
The manager of the Inlet Square Mall, near Ft.
Myers, Florida, wants to estimate the mean
amount spent per shopping visit by
customers. A sample of 20 customers
reveals the following amounts spent.
Confidence Interval Estimates for the Mean –
Calculated
Mean and Standard Deviation
302
Confidence Interval Estimates for the Mean –
By Formula
$60.
be
o
unlikely t
is
mean
population
that the
conclude
we
Hence,
interval.
confidence
in the
not
is
$60
of
value
The
$50.
be
could
mean
population
that the
reasonable
is
It
:
Conclude
$53.57
and
$45.13
are
interval
confidence
the
of
endpoints
The
22
.
4
35
.
49
20
01
.
9
093
.
2
35
.
49
20
01
.
9
35
.
49
unknown)
is
(since
dist.

t
the
using
C.I.
the
Compute
19
,
025
.
1
20
,
2
/
05
.
1
,
2
/
r
!
r
!
r
!
r
!
r
0
0
t
n
s
t
X
n
s
t
X
n
D
V
303
Confidence Interval Estimates for the Mean –
Using Excel
304
A Confidence Interval for a Proportion (
ʌ
)
The examples below illustrate the nominal scale of measurement.
1.
The career services director at Southern Technical Institute
reports that 80 percent of its graduates enter the job market
in a position related to their field of study.
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 Spring '11
 Leany
 Normal Distribution, Standard Deviation, pilot study, American Management Association