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Example:
In a recent poll, 70% of 1501 randomly
selected adults said they believed in global
warming.
Q: What is the proportion of the adult
population that believe in global warming?
Notation: p is the population proportion
(an unknown parameter).
p̂ is the sample proportion (computed).
From the poll data p̂ = 0.70.
Apparently, 0.70 will be the best estimate of
the proportion of all adults who believe in
global warming.
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1
Definition
A point estimate is a single value
(or point) used to approximate a
population parameter.
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2
Definition
ˆ
The sample proportion p is
the best point estimate of
the population proportion p.
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3
Example (continued)
We say that 0.70, or 70% is be the best point
estimate of the proportion of all adults who
believe in global warming.
But how reliable (accurate) is this estimate?
We will see that its margin of error is 2.3%.
This means the true proportion of adults who
believe in global warming is between 67.7%
and 72.3%. This gives an interval (from 67.7%
to 72.3%) containing the true (but unknown)
value of the population proportion.
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4
Definition
A confidence interval (or interval
estimate) is a range (or an interval)
of values used to estimate the true
value of a population parameter.
A confidence interval is sometimes
abbreviated as CI.
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5
Definition
A confidence level is the probability 1 –  (often
expressed as the equivalent percentage value)
that the confidence interval actually does contain
the population parameter.
The confidence level is also called degree of
confidence, or the confidence coefficient.
Most common choices are 90%, 95%, or 99%.
( = 10%), ( = 5%), ( = 1%)
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6
Example (continued)
In a recent poll, 70% of 1501 randomly
selected adults said they believed in global
warming.
The sample proportion p̂ = 0.70 is the best
estimate of the population proportion p.
A 95% confidence interval for the unknown
population parameter is
0.677 < p < 0.723
What does it mean, exactly?
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7
Interpreting a Confidence Interval
We are 95% confident that the interval from
0.677 to 0.723 actually does contain the true
value of the population proportion p.
This means that if we were to select many
different samples of size 1501 and construct
the corresponding confidence intervals, then
95% of them would actually contain the value
of the population proportion p.
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8
Caution
Know the correct interpretation of a
confidence interval.
It is wrong to say “the probability that
the population parameter belongs to
the confidence interval is 95%”
because the population parameter is
not a random variable, it does not
change its value.
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9
Caution
Do not confuse two percentages: the
proportion may be represented by
percents (like 70% in the example), and
the confidence level may be represented
by percents (like 95% in the example).
Proportion may be any number from 0%
to 100%.
Confidence level is usually 90% or 95%
or 99%.
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10
Definition
Margin of error, denoted by E, is the maximum
likely difference (with probability 1 – , such as
0.95) between the observed proportion p̂ and
the true value of the population proportion p.
The margin of error E is also called the
maximum error of the estimate.
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Confidence Interval for
a Population Proportion p
ˆ – E < p < pˆ + E
p
ˆ + E
p
ˆ + E)
(pˆ – E, p
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12
Finding the Point Estimate and E
from a Confidence Interval
ˆ
(upper confidence limit) + (lower confidence limit)
Point estimate of p:
ˆ
p=
2
Margin of Error:
E = (upper confidence limit) — (lower confidence limit)
2
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13
Next we learn how to construct
confidence intervals
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14
Critical Values
A z score can be used to distinguish between
sample statistics that are likely to occur and
those that are unlikely to occur. Such a z score
is called a critical value.
The standard normal distribution is divided into
three regions: middle part has area 1- and
two tails (left and right) have area /2 each:
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15
Critical Values
The z scores separate the middle interval (likely
values) from the tails (unlikely values). They
are z/2 and – z/2 , found from Table A-2.
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Definition
A critical value is the number on
the borderline separating sample
statistics that are likely to occur
from those that are unlikely to
occur.
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17
Notation for Critical Value
The critical value z/2 separates an
area of /2 in the right tail of the
standard normal distribution. The
value of –z/2 separates an area of
/2 in the left tail.
The subscript /2 is simply a
reminder that the z score separates
an area of /2 in the tail.
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18
Finding z2 for a 95%
Confidence Level
 = 5%
 2 = 2.5% = .025
z2
-z2
Critical Values
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19
Definition
Margin of error, denoted by E, is the maximum
likely difference (with probability 1 – , such as
0.95) between the observed proportion p̂ and
the true value of the population proportion p.
The margin of error E is also called the
maximum error of the estimate.
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20
Margin of Error for
Proportions
E  z 2
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ˆˆ
pq
n
21
Notation
E = margin of error
^ = sample proportion
p
q^ = 1 – ^
p
n = number of sample values
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Confidence Interval for
a Population Proportion p
pˆ – E < p < p̂ + E
where
E  z 2
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ˆˆ
pq
n
23
Confidence Interval for
a Population Proportion p
ˆ – E < p < pˆ + E
p
ˆ + E
p
ˆ + E)
(pˆ – E, p
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24
Finding the Point Estimate and E
from a Confidence Interval
ˆ
(upper confidence limit) + (lower confidence limit)
Point estimate of p:
ˆ
p=
2
Margin of Error:
E = (upper confidence limit) — (lower confidence limit)
2
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25
Round-Off Rule for
Confidence Interval Estimates of p
Round the confidence interval limits
for p to
three significant digits.
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26
Confidence Intervals by TI-83/84
•
•
•
•
•
•
•
•
•
Press STAT and select TESTS
Scroll down to 1-PropZInt
and press ENTER
Type in x: (number of successes)
n: (number of trials)
C-Level: (confidence level)
Press on Calculate
Read the confidence interval (…..,..…)
and the point estimate ^
p=…
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27
Sample Size
Suppose we want to collect sample
data in order to estimate some
population proportion. The question is
how many sample items must be
obtained?
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28
Determining Sample Size
z  2
E=
p
ˆ qˆ
n
(solve for n by algebra)
n=
( Z  2)2 p
ˆ ˆq
E2
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29
Sample Size for Estimating
Proportion p
ˆ
When an estimate of p is known:
n=
( z  2 )2 pˆ qˆ
E2
ˆ
When no estimate of p is known:
n=
( z 
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2
) 0.25
2
E2
30
Round-Off Rule for Determining
Sample Size
If the computed sample size n is not
a whole number, round the value of n
up to the next larger whole number.
Examples:
n=310.67
n=310.23
n=310.01
round up to 311
round up to 311
round up to 311
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31
Example:
A manager for E-Bay wants to determine the
current percentage of U.S. adults who now
use the Internet.
How many adults must be surveyed in order
to be 95% confident that the sample
percentage is in error by no more than three
percentage points?
a) In 2006, 73% of adults used the Internet.
b) No known possible value of the proportion.
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32
Example:
a) Use
pˆ  0.73 and qˆ  1  pˆ  0.27
  0.05 so z 2  1.96
E  0.03
z 

n
2
ˆˆ
pq
 2
E2
1.96   0.73  0.27 


2
 0.03 
2
 841.3104
 842
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To be 95% confident
that our sample
percentage is within
three percentage
points of the true
percentage for all
adults, we should
obtain a random
sample of 842 adults.
33
Example:
b) Use
  0.05 so z 2  1.96
E  0.03
z 

n
2
0.25
 2
E2
1.96  0.25


2
 0.03 
2
 1067.1111
 1068
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To be 95% confident that
our sample percentage
is within three
percentage points of the
true percentage for all
adults, we should obtain
a random sample of 1068
adults.
34
Section 7.3: Estimation of a
population mean m
s is known
In this section we cover methods for
estimating a population mean. In
addition to knowing the values of the
sample data or statistics, we must also
know the value of the population
standard deviation, s.
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35
Point Estimate of the
Population Mean
The sample mean x is the best point estimate
of the population mean µ.
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36
Confidence Interval for
Estimating a Population Mean
(with s Known)
m = population mean
s = population standard deviation
x = sample mean
n = number of sample values
E = margin of error
z/2 = z score separating an area of a/2 in the
right tail of the standard normal
distribution
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37
Requirements to check:
1. The value of the population standard
deviation s is known.
2. Either or both of these conditions is
satisfied: The population is normally
distributed or n > 30. (Just like in the
Central Limit Theorem.)
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38
Confidence Interval for
Estimating a Population Mean
(with s Known)
x  E  m  x  E where E  z 2 
or
x E
or
x  E,x  E 
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s
n
39
Definition
The two values x – E and x + E are
called confidence interval limits.
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40
Round-Off Rule for Confidence
Intervals Used to Estimate µ
1. When using the original set of data, round
the confidence interval limits to one more
decimal place than used in original set of
data.
2. When the original set of data is unknown
and only the summary statistics (n, x, s) are
used, round the confidence interval limits to
the same number of decimal places used for
the sample mean.
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41
Confidence Intervals by TI-83/84
•
•
•
•
•
•
•
•
•
Press STAT and select TESTS
Scroll down to ZInterval press ENTER
choose Data or Stats. For Stats:
Type in s: (known st. deviation)
_
x: (sample mean)
n: (sample size)
C-Level: (confidence level)
Press on Calculate
Read the confidence interval (…..,..…)
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42
Finding a Sample Size for
Estimating a Population Mean
m = population mean
σ = population standard deviation
x = population standard deviation
E = desired margin of error
zα/2 = z score separating an area of /2 in the right tail of
the standard normal distribution
n=
(z/2)  s
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2
E
43
Round-Off Rule for Sample Size n
If the computed sample size n
is not a whole number, round
the value of n up to the next
larger whole number.
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44
Example:
Assume that we want to estimate the mean IQ score for
the population of statistics students. How many
statistics students must be randomly selected for IQ
tests if we want 95% confidence that the sample mean
is within 3 IQ points of the population mean?
 = 0.05
 /2 = 0.025
z / 2 = 1.96
E = 3
s = 15
n =
1.96 • 15
2 = 96.04 = 97
3
With a simple random sample of only
97 statistics students, we will be 95%
confident that the sample mean is
within 3 IQ points of the true
population mean m.
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45
Section 7.4: Estimation of a
population mean m
s is not known
This section presents methods for
estimating a population mean when
the population standard deviation s
is not known.
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46
Sample Mean
_
The sample mean x is still
the best point estimate of
the population mean m.
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47
Construction of a confidence
intervals for m
s is not known
With σ unknown, we use the
Student t distribution instead
of normal distribution.
It involves a new feature:
number of degrees of freedom
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48
Definition
The number of degrees of freedom for a
collection of sample data is the number of
sample values that can vary after certain
restrictions have been imposed on all data
values.
The degree of freedom is often abbreviated df.
degrees of freedom = n – 1
in this section.
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49
Margin of Error E for Estimate of m
(With σ Not Known)
Formula 7-6
E = t 
s
2
n
where t2 has n – 1 degrees of freedom.
t/2 = critical t value separating an area of /2
in the right tail of the t distribution
Table A-3 lists values for tα/2
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50
Confidence Interval for the
Estimate of μ (With σ Not Known)
x–E <µ<x +E
where
E = t/2 s
n
df = n – 1
t/2 found in Table A-3
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51
Important Properties of the
Student t Distribution
1. The Student t distribution is different for different sample sizes
(see the following slide, for the cases n = 3 and n = 12).
2. The Student t distribution has the same general symmetric bell
shape as the standard normal distribution but it reflects the
greater variability (with wider distributions) than that the
standard normal distribution does.
3. The Student t distribution has a mean of t = 0 (just as the
standard normal distribution has a mean of z = 0).
4. The standard deviation of the Student t distribution varies with
the sample size and is greater than 1 (unlike the standard
normal distribution, which has a s = 1).
5. As the sample size n gets larger, the Student t distribution gets
closer to the normal distribution.
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52
Student t Distributions for
n = 3 and n = 12
Figure 7-5
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53
Choosing the Appropriate Distribution
Use the normal (z)
distribution
Use t distribution
s known and normally
distributed population
or
s known and n > 30
s not known and
normally distributed
population
or
s not known and n > 30
Methods of Chapter 7
do not apply
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Population is not
normally distributed
and n ≤ 30
54
Confidence Intervals by TI-83/84
• Press STAT and select TESTS
• Scroll down to TInterval press ENTER
• choose Data or Stats. For Stats:
_
• Type in x: (sample mean)

Sx: (sample st. deviation)
•
n: (number of trials)
•
C-Level: (confidence level)
• Press on Calculate
• Read the confidence interval (…..,..…)
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55
Confidence Intervals by TI-83/84
• Press STAT and select TESTS
• Scroll down to TInterval press ENTER
• choose Data or Stats. For Data:
• Type in List: L1 (or L2 or L3)
• (specify the list containing your data)

Freq: 1 (leave it)
•
C-Level: (confidence level)
• Press on Calculate
• Read the confidence interval (…..,..…)
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56
Finding the Point Estimate
and E from a Confidence Interval
Point estimate of µ:
x = (upper confidence limit) + (lower confidence limit)
2
Margin of Error:
E = (upper confidence limit) – (lower confidence limit)
2
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57
Section 7-5
Estimating a Population
Variance
This section covers the estimation
2
of a population variance s and
standard deviation s.
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58
Estimator of s
2
The sample variance s2 is the best
point estimate of the population
variance s2.
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59
Estimator of s
The sample standard deviation s is a
commonly used point estimate of s .
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60
Construction of confidence
2
intervals for s
We use the chi-square distribution,
denoted by Greek character 2
(pronounced chi-square).
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61
Properties of the Chi-Square
Distribution
1. The chi-square distribution is not symmetric, unlike
the normal and Student t distributions.
degrees of freedom = n – 1
Chi-Square Distribution
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Chi-Square Distribution for
df = 10 and df = 20
62
Properties of the Chi-Square
Distribution
2. The values of chi-square can be zero or positive, but
they cannot be negative.
3. The chi-square distribution is different for each
number of degrees of freedom, which is df = n – 1.
In Table A-4, each critical value of 2 corresponds to
an area given in the top row of the table, and that
area represents the cumulative area located to the
right of the critical value.
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63
Example
A sample of ten voltage levels is obtained.
Construction of a confidence interval for the
population standard deviation s requires the
left and right critical values of 2
corresponding to a confidence level of 95%
and a sample size of n = 10.
Find the critical value of 2 separating an area
of 0.025 in the left tail, and find the critical
value of 2 separating an area of 0.025 in the
right tail.
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64
Example
Critical Values of the Chi-Square Distribution
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65
Confidence Interval for Estimating a
Population Variance
n  1s

2
R
2
s
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2
n  1s



2
2
L
66
Confidence Interval for Estimating a
Population Standard Deviation
n  1s

2
R
2
s 
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n  1s

2
2
L
67
Requirement:
The population must have
normally distributed values
(even if the sample is large)
This requirement is very strict
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68
Round-Off Rule for Confidence
Intervals Used to Estimate s or s 2
1. When using the original set of data, round
the confidence interval limits to one more
decimal place than used in original set of
data.
2. When the original set of data is unknown
and only the summary statistics (n, x, s) are
used, round the confidence interval limits to
the same number of decimal places used for
the sample standard deviation.
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69
Determining Sample Sizes
The procedure for finding the sample size
necessary to estimate s2 is based on Table 7-2.
You just read the required sample size from an
appropriate line of the table.
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70
Determining Sample Sizes
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71
Example:
We want to estimate the standard deviation s.
We want to be 95% confident that our estimate
is within 20% of the true value of s.
How large should the sample be?
Assume that the population is normally
distributed.
From Table 7-2, we can see that 95% confidence
and an error of 20% for s correspond to a
sample of size 48.
We should obtain a sample of 48 values.
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72