powerpoint biostat
TRANSCRIPT
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Chapter Topics Estimation process
Point estimates
Interval estimates Confidence interval estimation for the
mean ( known)
Determining sample size Confidence interval estimation for the
mean ( unknown)
Confidence interval estimation for theproportion
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Estimation Process
Mean, , isunknown
Population Random Sample
MeanX = 50
Sample
I am 95%
confident that
is between 40& 60.
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Point EstimatesEstimate Population
Parameters with Sample
Statistics
Mean
Proportion
Variance
Difference
p
2
1 2
X
SP
2S
1 2X X
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Interval Estimates Provides range of values
Take into consideration variation in
sample statistics from sample tosample
Based on observation from 1 sample
Give information about closeness tounknown population parameters
Stated in terms of level of confidence
Never 100% sure
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Confidence Interval Estimates
Mean
Unknown
Confidence
Intervals
Proportion
Known
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Elements of
Confidence Interval Estimation Level of confidence
Confidence in which the interval will
contain the unknown populationparameter
Precision (range)
Closeness to the unknown parameter
Cost
Cost required to obtain a sample of
size n
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Level of Confidence Denoted by
A relative frequency interpretation
In the long run, of all theconfidence intervals that can beconstructed will contain the unknown
parameterA specific interval will either contain or
not contain the parameter
No probability involved in a specific interval
100 1 %
100 1 %
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Interval and Level of
Confidence
Confidence Intervals
Intervals
extend from
to of intervalsconstructedcontain ;
donot.
_Sampling Distribution of the Mean
XX Z
X
/ 2/ 2
XX
1
XX Z
100 1 %
100 %
/ 2 XZ
/ 2 XZ
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Confidence Interval for( Known)
Assumptions
Population standard deviation is
known Population is normally distributed
If population is not normal, use large
sample
Confidence interval estimate
/ 2 / 2X Z X Zn n
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Assumptions
Population standard deviation is unknown
Population is normally distributed
If population is not normal, use largesample
Use Students t Distribution Confidence Interval Estimate
Confidence Interval for( Unknown)
/ 2, 1 / 2, 1n n
S S
X t X tn n
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Degrees of Freedom (df) Number of observations that are free
to vary after sample mean has been
calculated Example
Mean of 3 numbers is 2
degrees of freedom= n -1
= 3 -1
= 2
1
2
3
1 (or any number)
2 (or any number)
3 (cannot vary)
X
X
X
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Studentst Table
Upper Tail Area
df .25 .10 .05
1 1.000 3.078 6.314
20.817 1.886
2.920
3 0.765 1.638 2.353
t0 2.920tValues
Let: n = 3
df = n - 1 = 2
= .10
/2 =.05
/ 2 = .05
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ExampleA random sample of 400 Voters showed 32preferred Candidate A. Set up a 95%
confidence interval estimate for p.
/ /
1 1
.08 1 .08 .08 1 .08.08 1.96 .08 1.96400 400
.053 .107
s s s s
s s
p p p pp Z p p Z
n n
p
p
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ExampleA random sample of 25 has 50 and 8.
Set up a 95% confidence interval estimate for
n X S
/ 2, 1 / 2, 1
8 850 2.0639 50 2.063925 25
46.69 53.30
n n
S SX t X t
n n
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Confidence IntervalEstimate for Proportion
Assumptions
Two categorical outcomes
Population follows binomial distribution
Normal approximation can be used ifand
Confidence interval estimate
5np 1 5n p
/ 2 / 2
1 1S S S SS S
p p p pp Z p p Z
n n