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Chapter 9
Section 2
Confidence Intervals about a
Population Mean in Practice where
the Population Standard Deviation
is Unknown
Chapter 9 – Section 2
●
Learning objectives
Know the properties of
t
distribution
Determine
t
values
Construct and interpret a confidence interval about a
population mean
1
2
3
Confidence Intervals
●
In Section 1, we assumed that we knew the
population standard deviation
σ
●
Since we did not know the population mean
μ
,
this seems to be unrealistic
●
In this section, we construct confidence intervals
in the case where we do not know the population
standard deviation
●
This is much more realistic
Confidence Intervals
●
If we don’t know the population standard
deviation σ, we obviously can’t use the formula
Confidence Intervals
●
Because we’ve changed our formula (by using
s
instead of
σ
), we can’t use the normal
distribution any more
●
Instead of the normal distribution, we use the
Student’s
t
distribution
●
This distribution was developed specifically for
the situation when
σ
is not known
Confidence Intervals
●
Properties of the
t
distribution
●
Several properties are familiar about the
Student’s
t
distribution
●
t
Just like the normal distribution, it is centered at 0 and
symmetric about 0
Just like the normal curve, the total area under the
Student’s
t
curve is 1, the area to left of 0 is ½, and
the area to the right of 0 is also ½
Just like the normal curve, as
t
increases, the
Student’s
t
curve gets close to, but never reaches, 0
Confidence Intervals
●
So what’s different?
●
Unlike the normal, there are many different
“standard”
t
distributions
There is a “standard” one with 1 degree of freedom
There is a “standard” one with 2 degrees of freedom
There is a “standard” one with 3 degrees of freedom
Etc.
●
The number of degrees of freedom is crucial for
the
t
distributions
Confidence Intervals
n
x
z
/
σ
μ

=
●
When
σ
is known, the
Z
score
follows a standard normal distribution
●
When
σ
is not known, the
t
statistic
follows a
t
distribution with
n
– 1
degrees
of
freedom
Helpful Hint: Review Example 1 on p. 467 of your textbook.
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