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Where we are going: a graphic:
Samples
1
2
Estimation (1 sample).
Paired
Means
Ho: / CI
Ho: / CI Ho: / CI
Variances
Ho: / CI
Ho: / CI
Proportions
Ho: / CI
Ho: / CI
Categories
Ho:
Ho:
Slopes
Ho: / CI
2 or more
Ho: / CI
1
One Sample: Estimation.
Gather data
population
sample
Make inferences and comparisons
parameters
, 2 , , , , etc .
statistics
2
ˆ, etc .

, 
ˆ, 
ˆ ,
ˆ, ˆ
or y , S2 S, p,
b, etc.
2
Estimation. The (1-) % Confidence Interval
mean
--is a bound, or interval which, in repeated sampling,
will cover the true mean (1- )% of the time. Sort of like
horseshoes, hand grenades, or fishing with a net.
Formula: mean
2
S
(1 - ) % CI  y  t
,(n 1) n
2
Where:  is the probability that the interval will be incorrect.
‘t’ is like z except appropriate for small samples
the subscripts for t tell how to look it up in the
table (appendix table 2 on page 1093 or the web page). 3
Pulse rate example: There were 128 measurements on
15 second pulse rate
the mean was 19.60 and
the standard deviation was 2.24.
mean
t
2
,(n 1)
 t.025,127  1.96 when   .05
So, a 95% CI is:
19.60 ± 1.96(2.24/11.31371)
= 19.60 ± 0.39
= 19.21
<<
19.99
we are 95% certain that this interval covers the true mean.
4
Estimation. The (1-) % Confidence Interval
variance
--is a bound, or interval which, in repeated sampling, will
cover the true variance (1- )% of the time. Same as for
the mean.
Formula: variance
(1 - )% CI 
2
(n - 1)S
2
 
 
 ,(n 1)
2
2
 
2
(n - 1)S
2
 


1 ,(n 1)
 2
Where:  is the probability that the interval will be incorrect.
‘2’ is a continuous distribution based on t2. See
appendix table 7 on page 1100 (or the web page).
5
Pulse rate example: variance : n=128 mean=19.6 s2=5.0312
 2

 1 -  ,(n 1 )
 2

2
 
  ,( n 1 )
2
  2.975 ,( 127 )  91.57 when   .05

2
.025,(127)
 152.21 when   .05
(127)5.0132
(127)5.0132
 2 
152 .21
91.57
So, a 95% CI is:
or

4.18   2  6.95

2.05  
 2.64
we are 95% certain that this interval covers the true variance
6
or standard deviation.
Estimation. The (1-) % Confidence Interval
proportion
--is a bound, or interval which, in repeated sampling,
will cover the true proportion (1- ) % of the time. Again,
just like the mean (large samples only).
Formula: proportion
p(1 - p)
(1 - )%CI  p  t
,(n 1)
n
2
Where:  is the probability that the interval will be incorrect.
‘t’ = z since the sample size is large.
7
Pulse rate example: proportion : we might wonder what
proportion of people have a pulse rate 18 or less.
P = 36/128 = .28
t
2
,(n 1)
 t.025,127  1.96 when   .05
So, a 95% CI is:
.28  1.96
.28(.72)
128
 .28  .08
 .20    .36
we are 95% certain that this interval covers the true mean.
The ‘.08’ is often called the ‘margin of error’.
8
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