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Chapter 12
Estimating
Means
with
Confidence
Copyright ©2011 Brooks/Cole, Cengage Learning
11.1 Introduction to CI for Means
• A parameter is a population characteristic – value
is usually unknown. We estimate the parameter using
sample information.
• A statistic, or estimate, is a characteristic of a sample.
A statistic estimates a parameter.
• A confidence interval is an interval of values
computed from sample data that is likely to include
the true population value.
• The confidence level for an interval describes our
confidence in the procedure we used. We are confident
that most of the confidence intervals we compute using
a procedure will contain the true population value.
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More Estimation Situations
Situation 1. Estimating the mean of a quantitative variable.
Example research questions:
What is the mean time that college students watch TV
per day? What is the mean pulse rate of women?
Population parameter: m (spelled “mu” and pronounced
“mew”) = population mean for the variable
Sample estimate: x = the sample mean for the variable
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More Estimation Situations
Situation 2. Estimating the population mean
of paired differences for a quantitative variable.
Example research questions:
What is the mean difference in weights for freshmen at the
beginning and end of the first semester? What is the mean
difference in age between husbands and wives in Britain?
Population parameter: md = population mean of the
differences in the two measurements
Sample estimate: d = the mean of the differences for a
sample of the two measurements
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More Estimation Situations
Situation 3. Estimating the difference between two
populations with regard to the mean
of a quantitative variable.
Example research questions:
How much difference is there in average weight loss for
those who diet compared to those who exercise to lose
weight? How much difference is there between the mean
foot lengths of men and women?
Population parameter: m1 – m2 = difference between the
two population means.
Sample estimate: x1  x2 = difference between the two
sample means.
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Paired Data
Paired data (or paired samples): when pairs of variables
are collected. Only interested in population (and sample)
of differences, and not in the original data.
• Each person measured twice. Two measurements of same
characteristic or trait are made under different conditions.
• Similar individuals are paired prior to an experiment. Each
member of a pair receives a different treatment.
Same response variable is measured for all individuals.
• Two different variables are measured for each individual.
Interested in amount of difference between two variables.
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Independent Samples
Two samples are called independent samples when
the measurements in one sample are not related to
the measurements in the other sample.
• Random samples taken separately from two
populations and same response variable is recorded.
• One random sample taken and a variable recorded, but
units are categorized to form two populations.
• Participants randomly assigned to one of two treatment
conditions, and same response variable is recorded.
• Two random samples taken from a population and a
separate variable is measured in each sample..
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Standard Errors
Rough Definition: The standard error of a
sample statistic measures, roughly, the average
difference between the statistic and the population
parameter. This “average difference” is over all
possible random samples of a given size that can
be taken from the population.
Technical Definition: The standard error of a
sample statistic is the estimated standard deviation
of the sampling distribution for the statistic.
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Standard Errors of a Sample Means
s
s.e.( x ) 
, s  sample standard deviation
n
sd
s.e.(d ) 
, sd  sample std dev of paired diffs
n
Example 11.2 Mean Hours Watching TV
Poll: Class of 175 students. In a typical day, about
how much time to you spend watching television?
Variable N Mean Median TrMean StDev
TV
175 2.09 2.000 1.950 1.644
SE Mean
0.124
s
1.644
s.e.x  

 .124
n
175
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Standard Error of the Difference
Between Two Sample Means
s.e.( x1  x2 ) 
s12 s22

n1 n2
Example 11.3 Lose More Weight by Diet or Exercise?
Study: n1 = 42 men on diet, n2 = 27 men on exercise routine
Diet: Lost an average of 7.2 kg with std dev of 3.7 kg
Exercise: Lost an average of 4.0 kg with std dev of 3.9 kg
So, x1  x2  7.2  4.0  3.2 kg
and s.e.( x1  x2 ) 
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3.7 2  3.92
42
47
 0.81
10
Using Student’s t-Distribution
to Determine the Multiplier
t-distribution: bell shape, centered at 0, more spread out
than standard normal curve from using s, instead of s.
t* multiplier: value in a t-distribution with
df = n – 1 such that the area between
-t* and +t* equals the desired confidence level.
Copyright ©2011 Brooks/Cole, Cengage Learning
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11.2 CI Module 3: Confidence
Intervals for One Population Mean
Lesson 1: Finding a CI for a Mean
for Any Sample Size and Any Confidence Level
Sample estimate  Multiplier  Standard error
Sample estimate: x = sample mean
s
s.e.x  
n
Standard error:
t-interval:
s
x t 
where df = n – 1 for the multiplier t*.
n
*
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Conditions Required
Two Situations t-interval is Valid:
Situation 1: The population of the measurements is
|bell-shaped, and a random sample of any size is measured.
In practice, for small samples, the data should show no
extreme skewness and should not contain any outliers.
Situation 2: The population of measurements is not
bell-shaped, but a large random sample is measured.
A somewhat arbitrary definition of a “large” sample
is n ≥ 30, but if there are extreme outliers, it is better to
have an even larger sample size than n = 30.
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Example 11.5 Mean Forearm Length
Data: Forearm lengths (cm)
for a random sample of n = 9 men
25.5, 24.0, 26.5, 25.5, 28.0, 27.0, 23.0, 25.0, 25.0
Note: Dotplot shows no obvious skewness and no outliers.
s
1.52
x  25.5, s  1.52, and s.e.x  

 .507
n
9
Multiplier t* from Table A.2 with df = 8 is t* = 2.31
95% Confidence Interval:
25.5  2.31(.507)  25.5  1.17  24.33 to 26.67 cm
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Calculating a Confidence Interval
for a Population Mean
t-interval:
s
x  t  s.e.x   x  t 
n
*
*
1.Make sure appropriate conditions apply checking sample size
and/or a shape picture. For small samples (n < 30) with extreme
skewness or outliers, you cannot proceed.
2.Choose a confidence level.
3.Compute the sample mean and standard deviation.
4.Calculate the standard error of the mean.
5.Calculate df = n – 1
6.Use Table A.2 (or software) to find the multiplier t*.
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Example 11.6 Watching TV
Poll: Class of 175 students. In a typical day, about
how much time to you spend watching television?
The sample mean was 2.09 hours and the sample standard
s
1.644
deviation was 1.644 hours.
s.e.x  

 .124
n
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Example 11.6 Watching TV
A 95% Confidence Interval:
2.09  1.98(.124)  2.09  .25  1.84 to 2.34 hours
A 99% Confidence Interval:
2.09  2.63(.124)  2.09  .33  1.76 to 2.42 hours
Note: the 99% CI estimate is wider than the 95% one.
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Example 11.6 Watching TV
Correct Interpretation of a CI for a Mean
We don’t know whether or not the mean TV viewing hours
for Penn State students is really between 1.84 to 2.34 hours.
•It is if we got one of the 95% of possible samples for which
the sample mean is close enough to the true population mean
for our procedure to work.
•It isn’t if we happened to get one of the 5% of possible
samples for which the sample mean is too far away from the
population mean for the procedure to work.
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Example 11.6 Watching TV
A Common Misinterpretation of a CI for a Mean
•We can be fairly certain that the population mean
forearm length is covered by this interval.
•This does not give us any information about the
range of individual forearm lengths.
•Do not fall into the trap of thinking that 95% of
men’s forearm lengths are in this interval.
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Example 11.7 What Students Sleep More?
Q: How many hours of sleep did you get last night,
to the nearest half hour?
Class
N
Stat 10 (stat literacy) 25
Stat 13 (stat methods) 148
Mean StDev SE Mean
7.66 1.34 0.27
6.81 1.73 0.14
Note: Bell-shape was reasonable for Stat 10 (with smaller n).
Notes: Interval for Stat 10 is wider (smaller sample size)
Two intervals do not overlap  Stat 10 average
significantly higher than Stat 13 average.
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Lesson 2: Special Case: Approximate
95% Confidence Intervals for Large Samples
For sufficiently large samples, the interval
Sample estimate  2  Standard error
is an approximate 95% confidence interval
for a population parameter.
Note: The 95% confidence level describes how often
the procedure provides an interval that includes the
population value. For about 95% of all random samples
of a specific size from a population, the confidence
interval captures the population parameter.
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Example 11.8 Watching TV
t-Interval:
2.09  1.98(.124)  2.09  .25  1.84 to 2.34 hours
Approximate 95% Confidence Interval:
2.09  2(.124)  2.09  .248  1.842 to 2.338 hours
Article: with 95% confidence, the mean is between
1.8 and 2.3 hours.
Technically: We are 95% confident the interval from
1.842 to 2.338 hours is correct in that it captures the
mean television viewing hours for Penn State students.
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11.3 CI Module 4: CI for Population
Mean of Paired Differences
Data: two variables for each of the n individuals or
pairs, use the difference d = x1 – x2
Population parameter: md = mean of the differences for
the population
Sample estimate: d = sample mean of the differences
Standard error: s.e.d  
sd
n
CI for md: d  t *  s.e.d  where df = n – 1 for the multiplier t*
Copyright ©2011 Brooks/Cole, Cengage Learning
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11.3 CI Module 4: CI for Population
Mean of Paired Differences
Data: two variables for n individuals or pairs;
use the difference d = x1 – x2.
Population parameter: md = mean of differences
for the population = m1 – m2.
Sample estimate: d = sample mean of the differences
Standard deviation and standard error:
sd
sd = std dev of sample of differences; s.e.d  
n
Confidence interval for md: d  t *  s.e. d
where df = n – 1 for the multiplier t*.
Copyright ©2011 Brooks/Cole, Cengage Learning
 
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Conditions Required
Two Situations t-interval is Valid:
Situation 1: The population of differences is |bell-shaped,
and a random sample of any size is measured. In practice,
for small samples, the differences in the sample should show
no extreme skewness and should not contain any outliers.
Situation 2: The population of differences is not bell-shaped,
but a large random sample is measured. A somewhat
arbitrary definition of a “large” sample is n ≥ 30 pairs,
but if there are extreme outliers in the sample of differences,
it is better to have an even larger sample size than n = 30.
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Example 11.9 Screen Time: Computer vs TV
Data: Hours spent watching TV and hours spent on
computer per week for n = 25 students.
Task: Make a 90% CI
for the mean difference
in hours spent using
computer versus
watching TV.
Note: Boxplot shows no obvious skewness and no outliers.
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Example 11.9 Screen Time: Computer vs TV
Results:
sd 15.24
d  5.36, sd  15.24, and s.e.d  

 3.05
n
25
Multiplier t* from Table A.2 with df = 24 is t* = 1.71
90% Confidence Interval:
5.36  1.71(3.05)  5.36  5.22  0.14 to 10.58 hours
Interpretation: We are 90% confident that the average
difference between computer usage and television viewing
for students represented by this sample is covered by the
interval from 0.14 to 10.58 hours per week, with more hours
spent on computer usage than on television viewing.
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Calculating a Confidence Interval for a
Population Mean of Paired Differences
sd
d  t  s.e.d   d  t 
n
*
*
1.Make sure appropriate conditions apply checking sample size
and/or a shape picture of the differences.
2.Choose a confidence level.
3.Compute the differences for the n pairs in the sample,
then find the mean and std dev for the differences.
4.Calculate the standard error of the mean difference.
5.Calculate df = n – 1
6.Use Table A.2 (or software) to find the multiplier t*.
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11.4 CI Module 5:
CI for Difference in Two Population
Means (Independent Samples)
Lesson 1: The General (Unpooled) Case
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CI for Difference for the Difference
between Two Population Means (Indep)
Approximate CI for m1 – m2:
x1  x2  t *
s12 s22

n1 n2
where t* is the value in a t-distribution with area
between -t* and t* equal to the desired confidence
level. Approximate df difficult to specify.
Use computer software or conservatively use the
smaller of the two sample sizes and subtract 1.
Note: for an approximate 95% CI, use a multiplier of 2.
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Degrees of Freedom
The t-distribution is only approximately correct
and df formula is complicated (Welch’s approx):
Statistical software can use the above
approximation, but if done by-hand then use a
conservative df = smaller of n1 – 1 and n2 – 1.
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Necessary Conditions
Two samples must be independent and either:
Situation 1: Populations of measurements both
bell-shaped, and random samples of any size
are measured.
Situation 2: Large (n  30) random samples are
measured. But if there are extreme outliers, or
extreme skewness, it is better to have an even
larger sample than n = 30.
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Example 11.11 Effect of a Stare on Driving
Randomized experiment: Researchers either stared
or did not stare at drivers stopped at a campus stop
sign; Timed how long (sec) it took driver to proceed
from sign to a mark on other side of the intersection.
No Stare Group (n = 14):
8.3, 5.5, 6.0, 8.1, 8.8, 7.5, 7.8,
7.1, 5.7, 6.5, 4.7, 6.9, 5.2, 4.7
Stare Group (n = 13):
5.6, 5.0, 5.7, 6.3, 6.5, 5.8, 4.5,
6.1, 4.8, 4.9, 4.5, 7.2, 5.8
Task: Make a 95% CI for the difference between
the mean crossing times for the two populations
represented by these two independent samples.
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Example 11.11 Effect of a Stare on Driving
Checking Conditions:
Boxplots show …
• No outliers and no strong skewness.
• Crossing times in stare group generally faster and less variable.
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Example 11.11 Effect of a Stare on Driving
Note: The df = 21 was reported by the computer package
based on the Welch’s approximation formula.
The 95% confidence interval for the difference between
the population means is 0.14 seconds to 1.93 seconds .
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Lesson 2: Equal Variance Assumption
and the Pooled Standard Error
May be reasonable to assume the two populations
have equal population standard deviations, or
equivalently, equal population variances: s 12  s 22  s 2
Estimate of this variance based on the combined
or “pooled” data is called the pooled variance.
The square root of the pooled variance is called
the pooled standard deviation:
Pooled standard deviation s p 
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n1  1s12  n2  1s22
n1  n2  2
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Pooled Standard Error
Pooled s.e.( x1  x2 ) 
s 2p
n1

s 2p
n2
1 1
 s   
 n1 n2 
2
p
 sp
1 1

n1 n2
Note: Pooled df = (n1 – 1) + (n2 – 1) = (n1 + n2 – 2).
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Pooled Confidence Interval
Pooled CI for the Difference Between
Two Means (Independent Samples):


1
1
x1  x2  t  s p
 
n1 n2 

*
where t* is found using a t-distribution
with df = (n1 + n2 – 2) and
sp is the pooled standard deviation.
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Example 11.14 Male and Female Sleep Times
Q: How much difference is there between how long
female and male students slept the previous night?
Data: The 83 female and 65 male responses from
students in an intro stat class.
Task: Make a 95% CI for the difference between
the two population means sleep hours for
females versus males.
Note: We will assume equal population variances.
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Example 11.14 Male and Female Sleep Times
Two-sample T for sleep [with “Assume Equal Variance” option]
Sex
Female
Male
N
83
65
Mean
7.02
6.55
StDev
1.75
1.68
SE Mean
0.19
0.21
Difference = mu (Female) – mu (Male)
Estimate for difference: 0.461
95% CI for difference: (-0.103, 1.025)
T-Test of difference = 0 (vs not =): T-Value = 1.62 P = 0.108 DF = 146
Both use Pooled StDev = 1.72
Notes:
• Two sample standard deviations are very similar.
• Sample mean for females higher than for males.
• 95% confidence interval contains 0 so cannot rule out
that the population means may be equal.
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Example 11.14 Male and Female Sleep Times
Pooled standard deviation and
pooled standard error “by-hand”:
Pooled std dev s p 

n1  1s12  n2  1s22
n1  n2  2
83  11.752  65  11.682
83  65  2
 2.957  1.72
Pooled s.e.( x1  x2 )  s p
 1.72
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1 1

n1 n2
1
1

 0.285
83 65
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Pooled or Unpooled?
• If the larger sample size produced the larger standard
deviation, the pooled procedure is acceptable because
it will be conservative.
• If the smaller standard deviation accompanies the
larger sample size, the pooled test can be quite
misleading and not recommended.
• If sample sizes are equal, the pooled and unpooled
standard errors are equal. Unless the sample standard
deviations are quite similar, it is best to use the
unpooled procedure.
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Example 11.15 Male and Female Sleep Times
Note: Unpooled s.e. is 0.30 while pooled s.e. is 0.36 (larger).
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Confidence Interval for the
Difference in Two Population Means
x1  x2   t *  s.e.x1  x2 
1.Make sure appropriate conditions apply checking sample size
and/or a shape picture of the differences.
2.Choose a confidence level.
3.Compute the mean and std dev for each sample.
4.Determine whether the std devs are similar enough to pooled
procedure can be used.
5.Calculate the appropriate standard error (pooled or unpooled).
6.Calculate the appropriate df.
7.Use Table A.2 (or software) to find the multiplier t*.
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