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Section 8.2
Estimating  When  is Unknown
In order to use the normal distribution to find the
confidence intervals for a population mean μ, we need to
x
know the value of σ, the population
standard deviation.
However much of the time, when μ is unknown, σ is
unknown as well. In such cases, we use the sample
standard deviation s to approximate σ. When we use s to
approximate σ, the sampling distribution for x
follows a new distribution called a
Student’s t distribution.
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Student’s t Distributions
The student’s t distribution uses a variable t defined as
follows. A student’s t distribution depends on sample
size n.
Assume that x has a normal distribution with mean μ.
For samples of size n with sample mean x and
standard deviation s, the t variable
x


has a Student’s t distribution with
t
s
degree of freedom d.f. = n – 1.
n
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The shape of the t distribution depends only on the
sample size, n, if the basic variable x has a normal
distribution.
When using the t distribution, we will assume that
the x distribution is normal.
Appendix Table 4 (Page A8)
gives values of the variable t corresponding to the
number of degrees of freedom (d.f.)
d.f. = n – 1
where n = sample size
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The t Distribution has a Shape
Similar to that of the the Normal
Distribution
4
Properties of a Student’s t Distribution
1.The distribution is Symmetric about the mean 0.
2.The distribution depends on the degrees of
freedom (d.f. = b-1 for μ confidence intervals.
3.The distribution is Bell-shaped with thicker tails
than the standard normal distribution.
4.As the degrees of freedom increase, the t
distribution approaches the standard normal
distribution
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Using Table 4 to find Critical Values for
confidence Intervals
Table 4 of the Appendix gives various t values for different
degrees of freedom d.f. We will use the table to find the
critical values tc for a c confidence level. In other words,
we want to find tc such thatt the area equal to c under the
t distribution for a given number of degrees of freedom
falls between -tc and tc
In the language of probability, we want to find tc such that
c
P(tc  t  tc )  c
This probability corresponds to the shaded area in next
figure
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Using Table 4 to Find Critical Values tc for a
c Confidence Level
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Using Table 4 to Find Critical Values of tc
Find the column in the table with the given c heading
Compute the number of degrees of freedom: d.f. = n  1
Read down the column under the appropriate c value
until we reach the row headed by the appropriate d.f.
If the required d.f. are not in the table:
Use the closest d.f. that is smaller than the needed d.f.
This results in a larger critical value tc.
The resulting confidence interval will be longer and
have a probability slightly higher than c.
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Example:
Find the critical value tc for a 95% confidence level
for a t distribution with sample size n = 8.
• Find the column in the table with c heading
0.950
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Example cont.
• Compute the number of degrees of freedom:
d.f. = n  1 = 8  1 = 7
Read down the column under the appropriate c value
until we reach the row headed by d.f. = 7
10
Example cont.
tc = 2.365
11
PROCEDURE
How to find Confidence Intervals
for  When  is Unknown
x E   x E
E  tc
s
n
c = confidence level (0 < c < 1)
tc = critical value for confidence level c, and
degrees of freedom d.f. = n  1
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Example
Confidence Intervals for ,  is Unknown
The mean weight of eight fish caught in a local lake is 15.7
ounces with a standard deviation of 2.3 ounces.
Construct a 90% confidence interval for the mean
weight of the population of fish in the lake.
Solution
Mean = 15.7 ounces
Standard deviation = 2.3 ounces.
n = 8, so d.f. = n – 1 = 7
For c = 0.90, Appendix Table 4 gives t0.90 = 1.895.
s
2.3
E  tc
 1.895
 1.54
n
8
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Example cont.
E = 1.54
The 90% confidence interval is:
15.7 - 1.54 <  < 15.7 + 1.54
14.16 <  < 17.24
We are 90% sure that the true mean weight of the
fish in the lake is between 14.16 and 17.24 ounces.
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Summary:
Confidence intervals for the mean
Assume that you have a random sample of size n > 1 from
an x distribution and that you have computed x and c.
A confidence interval for μ is
x E   x E
Where E is the margin of error.
How do you find E?
It depends of how of how much you know about the x
distribution.
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Summary cont.
Situation 1 (most common)
You don’t know the population standard deviation σ. In
this situation, you use the t distribution with margin of
error E  zc s
and degree of freedom d.f. = n – 1
n
Although a t distribution can be used in many situations,
you need to observe some guidelines. If n is less than
30, x should have a distribution that is mound-shaped
and approximately symmetric. Its even better if the x
distribution is normal.
If n is 30 or more, the central limit theorem implies that
these restrictions can be relaxed
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Summary cont.
Situation 2 (almost never happens!)
You actually know the population value σ.
In addition, you know that x has a normal distribution.
If you don’t know that the x distribution is normal,
then your sample size n must be 30 or larger.
In this situation, you use the standard normal z distribution
with margin of error
E  zc

n
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Summary cont.
Which distribution should you use for
Examine problem statement x ?
(a) If σ is known
use normal distribution with margin of error

E  zc
n
(b) If σ is not known
use Student’s t distribution with margin of error
E  tc
Assignment 20
s
n
d.f. = n - 1
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