Author(s): Brenda Gunderson, Ph.D., 2011
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Mind on Statistics
Utts/Heckard, 3rd Edition, Duxbury, 2006
Text Only: ISBN 0495667161
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Stat 250 Gunderson Lecture Notes
Learning about a Population Mean Difference
Part 2: Confidence Interval for a Population Mean Difference
Chapter 11: Section 3, CI Module 4
11.3 CI Module 4: Confidence Interval for the
Population Mean of Paired Differences d
Recall that an important special case of a single mean of a population occurs when two
quantitative variables are collected in pairs, and we desire information about the difference
between the two variables. For paired data designs, it is the differences that we are interested
in analyzing. By focusing on the differences we again have just one sample of observations (the
differences) and are able to use the confidence interval for the population mean difference.
Notation:
Population Parameter: d = population mean difference
Data: d1 , d 2 ,..., d n
Sample Estimate: d = sample mean difference
Standard Error: s.e.( d ) =
sd
n
where sd is the standard deviation of the sampled differences.
We use the sample estimate and its standard error to form a confidence interval estimate for
the parameter using the following form:
Sample Estimate  Multiplier x Standard error
The multiplier used will depend on the confidence level, the sample size, and the type of
parameter being estimated. In this case, since we are estimating a single population mean, the
multiplier will be a t* value.
Here is the summary for a paired data confidence interval:
One-sample t Confidence Interval for the Population Mean Difference d
d  t *s.e.(d )
where t * is the appropriate value for a t(n – 1) distribution.
Note that n is the number of pairs, or the number of differences.
This interval requires that the differences can be considered a random sample from a normal
population. If the sample size is large, the assumption of normality is not so crucial and the
result is approximate.
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.
Used with permission.
139
Try It! Changes in Reasoning Scores
Do piano lessons improve spatial-temporal reasoning of preschool children?
Data: Change in reasoning score, after piano lessons - before piano lessons, with larger values
indicating better reasoning, for a random sample of n = 34 preschool children.
2
3
3
5
4
4
7
9
6
-2
4
7
2
5
-2
7
2
7
4
9
-3
1
6
3
0
0
4
7
3
4
3
6
4
-1
(a) Display the data, summarize the distribution. These data were entered into SPSS to
produce the following histogram.
Notes:
1. Diff = after – before so …
2. Sample mean difference = ______
3. Normality of the response
(the difference) for the population?
Change in Reasoning
Some summary measures:
One-Sample Statistics
N
REASCORE
34
Mean
3.62
Std. Deviation
3.06
Std. Error
Mean
.52
(b) Give a 95% confidence interval for the mean improvement in reasoning scores.
(c) What value is of particular interest to see whether or not it is in the interval?
(d) A student in your class wrote the following interpretation about the 95% confidence level
used in making the interval. Is it a correct interpretation? If not, update it to make it correct.
“If this study were repeated many times, we would expect 95% of the resulting
confidence intervals to contain the sample mean improvement in reasoning scores.”
140
SPSS Note:
The differences were already computed and entered as the data. So to make a confidence
interval with SPSS we would need to perform a one-sample t-test on the differences and specify
a test value of 0. We should also check the options box to be sure the confidence level is the
one you want, namely 95%.
One-Sample Test
Tes t Value = 0
REASCORE
t
6.904
df
33
Sig. (2-tailed)
.000
Mean
Difference
3.62
95% Confidence
Interval of the
Difference
Lower
Upper
2.55
4.68
If the before and after scores were entered into SPSS, then we would use a paired t-test option.
SPSS would compute the differences for us and provide the confidence interval results. Details
of the SPSS steps for analyzing paired data can be found in your Lab Workbook.
Below is the summary of confidence intervals for the big 5 parameters covered in Chapters 10
and 11. We have covered four of the five scenarios, those in the entire top half that are circled.
Population Proportion
Parameter
Parameter

x
Statistic
Standard Error
Parameter
pˆ 1  pˆ 2
Statistic
Standard Error
p
pˆ (1  pˆ )
n
s.e.( pˆ 1  pˆ 2 ) 
Confidence Interval
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
s.e.( x ) 
Conservative Conf. Interval
s
n
Confidence Interval
pˆ  z *s.e.( pˆ )
pˆ 
Population Mean
p1  p 2
p̂
Statistic
Standard Error
s.e.( pˆ ) 
Two Population Proportions
Confidence Interval
x  t *s.e.( x )
df = n – 1
 pˆ 1  pˆ 2   z *s.e. pˆ 1  pˆ 2 
z*
Paired Confidence Interval
d  t *s.e.(d )
df = n – 1
2 n
 z* 

Sample Size n  

2
m


2
Two Population Means
General
Parameter
Statistic
Standard Error
s.e.x1  x2  
s12 s22

n1 n2
1   2
x1  x 2
Pooled
x1  x 2
pooled s.e.x1  x2   s p
where s p 
Confidence Interval
1   2
Parameter
Statistic
Standard Error
(n1  1)s12  (n 2  1) s 22
n1  n 2  2
Confidence Interval
141
1 1

n1 n2
x1  x2   t * s.e.(x1  x2 )
df = min( n1  1, n2  1)
x1  x2   t * pooled s.e.(x1  x2 )
df = n1  n2  2
Additional Notes
A place to … jot down questions you may have and ask during office hours, take a few extra notes, write
out an extra practice problem or summary completed in lecture, create your own short summary about
this chapter.
142