R Programming
Confidence Intervals for
Population Means
Confidence Intervals
These notes will guide you through estimating a single
population mean from a sample.
Throughout these notes:
1. The formula will be presented;
2. The formula will be applied (manually);
3. The formula will be applied via R.
Confidence Intervals
Any Confidence Interval can be estimated using the
following general form:
Sample estimate + conf. level * standard error
A Confidence Interval around a single population
mean is developed using:
𝒙 ± 𝒛 ∗ 𝒔/ 𝒏
Where:
x = sample mean
z = the appropriate two sided Z-score, based upon the desired
confidence level
s = sample standard deviation
n = number of elements in sample
Confidence Intervals
Typical Z scores used in CI Estimation:
90% confidence = 1.645
95% confidence = 1.96
98% confidence = 2.33
99% confidence = 2.575
Confidence Intervals
For example, lets say that we took a poll of 100 college
students and determined that they spent an average of
$225 on books in a semester with a std dev of $50.
Report the 95% confidence interval for the expenditure
on books for ALL college students.
Confidence Intervals
In this example,
𝑥 = 225
z= 1.96
s = 50
n= 100
So, the 95% interval would be:
225 ± 1.96 ∗
= 225 ± 9.8
This becomes 225 + 9.8
50
100
Confidence Intervals
In English, this becomes…
“We are 95% confident that the mean expenditure
on books for college students is $225 plus or minus
$9.80…we are 95% confident that they are
spending as little as $215.20 or as much as
$234.80”.
Confidence Intervals
One general note regarding Confidence Intervals…
The results tell us NOTHING about the probability of an
individual observation…a 95% interval SHOULD NOT be
interpreted as “Joe has a 95% probability of spending
between $215.20 and $234.80”.
The interval is an estimation of the mean of the
population…not of an individual observation.
Confidence Intervals
A second example…
175 students from Penn State where asked “In a typical
day, about how much time do you spend watching TV?”
The sample average was 2.09 hours with a standard
deviation of 1.644.
Report the 99% confidence interval for the average time
spent watching TV for all college students.
Confidence Intervals
The answer is:
“We are 99% confident that the average time college
students spent watching TV is between 1.77 and 2.41
hours. “
R Programming
Confidence Intervals for The
Population Mean of Paired
Differences
Confidence Intervals
These notes will guide you through estimating Confidence
Intervals for the Population Mean of Paired Samples.
Throughout these notes:
1. The formula will be presented;
2. The formula will be applied (manually);
3. The formula will be applied via R.
Confidence Intervals
A few notes about paired differences (which are
different from two independent sample differences):
• The same (or VERY similar) people/objects are
measured pre/post treatment.
• Typically, we are only interested in the calculated
differences between the before and after - not in the
actual values of the original data which was collected.
Confidence Intervals
As we saw previously, any CI can be estimated using
the approach of
Sample estimate + conf. level * standard error
A Confidence Interval around the population mean of
paired differences :
𝒙𝒅 ± 𝒛 ∗ 𝒏
Where:
x d = sample mean (difference of the two means)
z = the appropriate two sided Z-score, based upon desired
confidence
s = sample standard deviation (difference)
n = number of elements in sample
Confidence Intervals
Typical Z scores used in CI Estimation:
90% confidence = 1.645
95% confidence = 1.96
98% confidence = 2.33
99% confidence = 2.575
90% of the area under the curve
-1.645
0
1.645
Confidence Intervals
For example, lets say that a particular firm tracks their
sales every week over the course of a year. They
average 150 units a week. After hiring an advertising
company, the average goes up to 165 units on average
the next year. The std of the differences between the
two years is 10.25. What is the 90% Confidence Interval?
Confidence Intervals
In this example,
xd = 15
z = 1.645
s = 10.25
n = 52
So, the 90% interval would be:
15+1.645*(10.25/SQRT(52))…
This becomes 15+ 2.3382
Week
Year 1
Year 2
Difference
1
100
125
25
2
160
180
20
3
110
150
40
4…
110…
120…
10…
Overall
Average
15
Overall
Std
10.25
Confidence Intervals
In English, this becomes…
“We are 90% confident that the mean difference
in weekly sales between year 1 and year 2 is 15
plus or minus 2.3382…we are 90% confident that
the change is as little as 12.6618 sales per week
and as great as 17.3382 sales per week.”
Confidence Intervals
Lets now put this into a more realistic context…
Say that each sale generates $100 in profit. If you
are generating as little as 12.6618 incremental
sales per week (lower end of the interval), that
would equate to $65,841.36 (12.6618*100*52)
incremental profit over one year. This might
represent the maximum that you would be willing
to pay the advertising agency.
Confidence Intervals
A second example…
A pharmaceutical company is testing a new cholesterol
reducing drug – Choless. They recruit 100 representative
men for the trial of Choless. The average cholesterol
level before the trial was 242. The average cholesterol
level after the trial was 216. The average difference was
26 with a standard deviation of the differences of 32.
Determine the 99% confidence level.
Confidence Intervals
In this example,
xd = 26
z = 2.575
s = 32
n = 100
So, the 99% interval of the paired differences would be:
26+2.575*(32/SQRT(100))…
This becomes 26+ 8.24
Confidence Intervals
In English, this becomes…
“We are 99% confident that the mean difference
in cholesterol levels before Choless and after
Choless is 26 points, plus or minus 8.24 points…we
are 99% confident that the change is as little as
17.76 points or as great as 34.24 points.”
R Programming
Confidence Intervals for
Difference The Two
Independent Sample Means
Confidence Intervals
These notes will guide you through estimating parameter
(mean) confidence intervals for two independent samples.
Throughout these notes:
1. The formula will be presented;
2. The formula will be applied (manually);
3. The formula will be applied via R.
Confidence Intervals
As we saw previously, any CI can be estimated using
the approach of
Sample estimate + conf. level * standard error
A Confidence Interval around the difference between
two independent samples can be calculated as:
x1 – x2  z* SQRT((s21/n1)+(s22/n2))
Where:
xi = sample mean (two independent samples)
z = the appropriate two sided Z-score, based upon desired
confidence
si = sample standard deviation (two independent samples)
ni = number of elements in each sample
Confidence Intervals
Typical Z scores used in CI Estimation:
90% confidence = 1.645
95% confidence = 1.96
98% confidence = 2.33
99% confidence = 2.575
Confidence Intervals
A few notes about independent sample differences:
• The two samples must be statistically independent of
each other.
• You need to know if the variances (std) are
approximately equal or not. The formula from the
previous slide assumes that they are not equal. A
second formula and a discussion of the differences will
be provided in a later slide.
Confidence Intervals
Volunteers who had developed a cold within the previous 24
hours were randomly assigned to two groups – one took zinc
lozenges and one took a placebo every 2-3 hours until their
symptoms had subsided.
From Zinc Group:
From Placebo Group:
35 people
Duration of symptoms –
4.5 days
Std of days – 1.6
33 people
Duration of symptoms –
8.1 days
Std of days – 1.8
Calculate the 95% Confidence Interval for the difference
between the two groups.
Confidence Intervals
In this example,
x1 = 4.5 x2 = 8.1
z= 1.96
s1 = 1.6 s2 = 1.8
n1= 35 n2 = 33
So, the 95% interval would be:
4.5 – 8.1  1.96* SQRT((1.62/35)+(1.82/33))…
This becomes -3.6 + .8112
Confidence Intervals
In English, this becomes…
“We are 95% confident that the zinc group
experienced 3.6 fewer days of symptoms than did
the placebo group, plus or minus .8112 days…the
zinc group experienced as much as 4.4112 days
less than the placebo group or as few as 2.7888
days less than the placebo group”.
Confidence Intervals
In this example, we generated the Confidence Interval
using the unpooled approach . There is a second option
– the pooled approach.
In theory, we use the pooled approach when the
standard deviations are approximately the same between
the two groups. In practice, this is uncommon.
The rule of thumb goes something like this…
If the larger sample standard deviation is from the group with the
larger sample size, the pooled procedure will generate a larger
(more conservative) interval. If the smaller sample standard
deviation is from the group with the larger sample size, the
pooled version may produce a misleading narrow interval.
Confidence Intervals
Practitioners tend to use the unpooled procedure unless
the sample standard deviations are VERY close.
Lets redo the previous example with the pooled
procedure and discuss the difference…
x1 – x2  z* SQRT(s2(1/n1+1/n2))
Where s = SQRT[((n1-1)s21+(n2-1)S22)/(n1+n2 – 2)]
So, the 95% interval would be:
4.5 – 8.1  1.96*1.699*[ SQRT( (1/35)+(1/33))]…
-3.6+.8080
Confidence Intervals
Using the pooled version, we did generate a slightly
smaller margin of error (.8080 versus .8112). This occurred
because the larger group (group 1) had smaller standard
deviation.
As mentioned in the previous slide, when in doubt, use the
unpooled approach.
Confidence Intervals
Lets now generate two sample Confidence Intervals
using R…
R Programming
Confidence Intervals for One
Sample Proportion
Confidence Intervals
These notes will guide you through estimating proportion
confidence intervals.
In each case:
1. The formula will be presented;
2. The formula will be applied (manually);
3. The formula will be applied via R.
Confidence Intervals
The interval for any CI estimate can be expressed as:
Sample estimate + conf. level * standard error
In the case of a single population proportion, the expression
is:
𝒑 ±𝒛∗
𝒑(𝟏 − 𝒑)
𝒏
Where, “p” is the proportion of units in a sample;
z is the associated # of Std deviations associated
with the required confidence level;
n is the number of obs in the sample.
Confidence Intervals
Typical Z scores used in CI Estimation:
90% confidence = 1.645
95% confidence = 1.96
98% confidence = 2.33
99% confidence = 2.575
Confidence Intervals
The Gallup Organization, founded in 1935 by George
Gallup, is one of the most well respected polling
organizations in the world.
Their website is a great place to find confidence intervals of
proportions www.gallup.com
A common survey for Gallup is the presidential approval
rating…
Confidence Intervals
Before we dissect a Gallup survey, lets take a look at their
methodology statement from their website:
Survey Methods
Results are based on telephone interviews with 997 national adults, aged 18 and older,
conducted May 19, 2009. For results based on the total sample of national adults, one can
say with 95% confidence that the maximum margin of sampling error is ±3 percentage
points.
Interviews are conducted with respondents on land-line telephones (for respondents with a
land-line telephone) and cellular phones (for respondents who are cell-phone only).
In addition to sampling error, question wording and practical difficulties in conducting
surveys can introduce error or bias into the findings of public opinion polls.
Polls conducted entirely in one day, such as this one, are subject to additional error or bias
not found in polls conducted over several days.
Confidence Intervals
From this statement, Gallup asked 997 if they approve of the
job the president is doing. From this representative sample,
64% say “Yes”…How did they get a margin of error of 3%?
So, from this example, we have:
p=.64
z = 1.96
n = 997
.64 + 1.96 * SQRT((.64(.36))/997) …
.64+.0298
Confidence Intervals
So…what would happen to the interval if we increased the
confidence to 99%?
Everything would stay the same except for the Z-score:
p=.64
z = 2.575
n = 997
.64 + 2.575 * SQRT((.64(.36))/997) …
.64+.0391
Confidence Intervals
From the Gallup Data (approval = 64%, n=997),
here are the effects of changing the confidence
levels…
Confidence
Level
Z-score
Margin of
Error
Low End of
Approval
Rating
High End of
Approval
Rating
90%
1.645
.0250
61.50%
66.50%
95%
1.96
.0298
61%
67%
99%
2.575
.0391
60%
68%
R Programming
Confidence Intervals for Difference
of Two Sample Proportions
Confidence Intervals
These notes will guide you through estimating the
confidence interval for the difference in proportions for two
independent samples.
Throughout these notes:
1. The formula will be presented;
2. The formula will be applied (manually);
3. The formula will be applied via R.
Confidence Intervals
As we saw previously, the interval for any CI estimate can
be expressed as:
Sample estimate + conf. level * standard error
In the case of a CI for the difference between two
proportions, the expression is:
𝒑𝟏 (𝟏 − 𝒑𝟏 ) 𝒑𝟐 (𝟏 − 𝒑𝟐 )
𝒑𝟏 − 𝒑𝟐 ± 𝒛 ∗
+
𝒏𝟏
𝒏𝟐
Where, pi is the proportion of units in a sample (1 or 2);
Z is the associated # of Std deviations associated
with the required confidence level;
ni is the number of obs in the sample (1 or 2).
Confidence Intervals
A few notes about independent sample differences:
• Sample proportions are available based upon
independent randomly selected samples from the two
populations.
• The frequencies of each proportional quantity is at
least 10.
Confidence Intervals
For example, lets say that we took a poll of students and
asked “would you date someone with a great
personality who you were not attracted to?” By gender,
the results were 61.07% of 131 women said “yes” while
42.62% of 61 men said “yes”.
What is the 95% Confidence Interval?
Confidence Intervals
So, from this example, we have:
p1=.6107 p2 = .4262
z = 1.96
n1 = 131 n2 = 61
.6107 –.4262 + 1.96 * SQRT(((.6107(.3893))/131)+((.4262(.5738)/61)))
.1850 + .1493
Confidence Intervals
In English, this becomes…
“We are 95% confident that the proportion of
women who would date someone who they
thought was a great person even if they did not
find them attractive is 18.50% higher than the
proportion of men…this difference could be as
great as 33.43% or as little as 3.57%.”
Confidence Intervals
Lets now generate a single proportion confidence
interval using R…