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127
Lecture 13:
CHAPTER 5: CONFIDENCE INTERVALS
SmallSample Confidence Intervals for a
Population Mean – Cont’d (Section 5.3, page 321)
Recall:
Last class we introduced the student’s tdistribution which is
used when the population is approximately normal and the sample
size is small.
n
s
x
t
follows the tdistribution with n1 degrees of freedom.
Confidence Intervals Using the Student’s t Distribution
If a random sample X
1
, X
2
. .
. X
n
is chosen from a normal
distribution; then the confidence Interval of
is
SE
t
x
df
)
,
2
(
Example:
The article “Direct StrutandTie Model for Prestressed
Deep Beams” presents measurements of the nominal shear strength
(in kN) for a sample of 15 prestressed concrete beams.
The results
are
580 400 428 825 850 875 920 550 575 750 636 360
590 735 950
Assume the data come from a normal distribution and there are no
outliers.
Construct a 99% confidence interval for the mean shear
strength.
(Note:
If it was not stated that the data follow a normal
distribution, we would need to find out if it follows a normal
distribution by either a histogram/boxplot because the sample is
small.
When the sample is small, we can not assume normality.
We
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need to be certain that there are no outliers and the distribution is
symmetric.)
129
Example:
For the data in the above example, if we wanted to find a
95% confidence interval instead of a 99% confidence interval, what
value in the formula would change?
What would the new CI
become?
Question:
Considering these 2 examples, if the level of the
confidence is decreased and all other things remain the same, the
width of the confidence interval will ________________.
Confidence Intervals for the Difference Between
Two Means (Section 5.4, page 354)
We now investigate examples in which we wish to estimate the
difference between the means of two populations.
The data will
consist of two samples, one from each population.
For this section,
we will only be dealing with large sample means.
Summary
Let
x
n
X
X
,...,
1
be a large random sample of size
x
n
from a
population with mean
x
and standard deviation
x
, and let
y
n
Y
Y
,...,
1
be a large random sample of size
y
n
from a population
with mean
y
and standard deviation
y
.
If the two samples are
independent, then a level
)%
1
(
100
confidence interval for
y
x
is
y
y
x
x
n
n
z
Y
X
2
2
2
)
(
When the values of
x
and
y
are unknown, they can be replaced
with the sample standard deviations
x
s
and
y
s
.
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Example:
The chemical composition of soil varies with depth.
Fifty
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This note was uploaded on 06/20/2011 for the course STAT 2800 taught by Professor Paula during the Winter '11 term at UOIT.
 Winter '11
 Paula
 TDistribution

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