Inference for Distributions
Inference for the Mean of a Population
Section 7.1
© 2009 W.H Freeman and Company
Sweetening colas
Cola manufacturers want to test how much the sweetness of a new
cola drink is affected by storage. The sweetness loss due to storage
was evaluated by 10 professional tasters (by comparing the sweetness
before and after storage):
Taster
Sweetness loss
!
1
2.0
!
2
0.4
!
3
0.7
!
4
2.0
!
5
!
0.4
!
6
2.2
!
7
!
1.3
!
8
1.2
!
9
1.1
!
10
2.3
Obviously, we want to test if
storage results in a loss of
sweetness, thus:
H
0
:
μ
= 0 versus
H
a
:
> 0
This looks familiar. However, here we do not know the population parameter
!
.
"
The population of all cola drinkers is too large.
"
Since this is a new cola recipe, we have no population data.
This situation is very common with real data.
When
is unknown
!
When the sample size is large,
the sample is likely to contain
elements representative of the
whole population. Then
s
is a
good estimate of
.
Population
distribution
Small sample
Large sample
!
But when the sample size is
small, the sample contains only
a few individuals. Then
s
is a
more mediocre estimate of
.
The sample standard deviation
s
provides an estimate of the population
standard deviation
.
A study examined the effect of a new medication on the seated
systolic blood pressure. The results, presented as mean ± SE for
25 patients, are 113.5 ± 8.9.
What is the standard deviation
s
of the sample data?
Standard error
For a sample of size
n
,
the sample standard deviation
s
is:
n
!
1
is the “degrees of freedom.”
The value
s
/
!
n
is called the standard error of the mean or
SE
.
Scientists often present sample results as mean ± SE.
SE =
s
/
"
n
<=>
s
= SE*
"
n
s
= 8.9*
"
25 = 44.5
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View Full DocumentThe
t
distributions
Suppose that an SRS of size
n
is drawn from an
N
(
"
,
#
) population.
!
When
!
is known, the sampling distribution is
N
(
μ
,
/
"
n
).
!
When
is estimated from the sample standard deviation
s
, the
sampling distribution follows a
t
distribution with degrees of
freedom
n
"
1.
is the
onesample
t
statistic.
When
n
is very large,
s
is a very good estimate of
and the
corresponding
t
distributions are very close to the normal distribution.
The
t
distributions become wider for smaller sample sizes, reflecting the
lack of precision in estimating
from
s
.
The onesample
t
confidence interval
The
level
C
confidence interval
is an interval with probability
C
of
containing the true population parameter.
We have a data set from a population with both
and
unknown. We
use
to estimate
, and
s
to estimate
,
using a
t
distribution, df = n
!
1.
C
t
*
!
t
*
m
m
Practical use of
t
:
t
*
!
C
is the area between
!
t
* and
t
*.
!
We find
t
* in the line of Table D
for df = n
!
1 and confidence level
C
.
!
The margin of error
m
is:
Red wine, in moderation
Drinking red wine in moderation may protect against heart attacks. The
polyphenols it contains act on blood cholesterol and thus are a likely cause.
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 Spring '08
 HOLT
 Normal Distribution, Standard Deviation

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