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Chapter 7
Confidence Intervals and
Sample Sizes
7.2 Estimating a Proportion p
7.3 Estimating a Mean µ (σ known)
7.4 Estimating a Mean µ (σ unknown)
7.5 Estimating a Standard Deviation σ
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Example 1
In a recent poll, 70% or 1501 randomly selected adults
said they believed in global warming.
Q: What is the proportion of the adult population
that believe in global warming?
TRICK QUESTION!
We only know the sample proportion s,
We do not know the population proportion σ.
BUT…
The proportion of the sample (0.7) is our
best point estimate (i.e. best guess).
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Definition
Point Estimate
A single value (or point) used to approximate
a population parameter
Best Point
Estimate
Population
Parameter
Proportion
p
≈
p
Mean
µ
≈
x
Std. Dev.
σ
≈
s
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Definition
Confidence Interval : CI
The range (or interval) of values to estimate
the true value of a population parameter.
It is abbreviated as CI
In Example 1, the 95% confidence interval for the
population proportion p is CI = (0.677, 0.723)
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Definition
Confidence Level : 1 – α
The probability that the confidence interval
actually contains the population parameter.
The most common confidence levels used
are 90%, 95%, 99%
90% : α=0.1
95% : α=0.05
99% : α=0.01
In Example 1, the Confidence level is 95%
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Definition
Margin of Error : E
The maximum likely difference between
the observed value and true value of the
population parameter (with probability is 1–α)
The margin of error is used to determine a
confidence interval (of a proportion or mean)
In Example 1, the 95% margin of error for the population
proportion p is E = 0.023
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Example 1 Continued…
In a recent poll, 70% or 1501 randomly selected
adults said they believed in global warming.
Q: What is the proportion of the adult
population that believe in global warming?
A: 0.7 is the best point estimate of the proportion
of all adults who believe in global warming.
The 95% confidence interval of the population
proportion p is:
CI = (0.677, 0.723)
( with a margin of error E = 0.023 )
What does it mean, exactly?
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Interpreting a Confidence Interval
For the 95% confidence interval CI = (0.677, 0.723)
we say:
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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!!! Caution !!!
Know the correct interpretation of a
confidence interval
It is incorrect to say
“ the probability that the population
parameter belongs to the confidence
interval is 95% ”
because the population parameter is not
a random variable, its value cannot change
The population is “set in stone”
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!!! Caution !!!
Do not confuse the two percentages
The proportion can be represented
by percents (like 70% in Example 1)
The confidence level may be represented
by percents (like 95% in Example 1)
Proportions can be any value from 0% to 100%
Confidence levels are usually 90%, 95%, or 99%
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Confidence Interval Formula
( y – E, y + E )
y = Best point estimate
E = Margin of Error
• Centered at the best point estimate
• Width is determined by E
The value of E depends the critical value of the CI
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Finding the Point Estimate and E
from a Confidence Interval
Point estimate : y
y = (upper confidence limit) + (lower confidence limit)
2
Margin of Error : E
E = (upper confidence limit) — (lower confidence limit)
2
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Definition
Critical Value
The number on the borderline separating
sample statistics that are likely to occur from
those that are unlikely to occur.
A critical value is dependent on a probability
distribution the parameter follows and the
confidence level (1 – α) .
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Normal Dist. Critical Values
For a population proportion p and mean µ
(σ known), the critical values are found using
z-scores on a standard normal distribution
The standard normal distribution is divided into
three regions: middle part has area 1 – α and
two tails (left and right) have area α/2 each:
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Normal Dist. Critical Values
The z-scores za/2 and –za/2 separate the values:
Likely values
( middle interval )
Unlikely values
( tails )
Use StatCrunch to calculate z-scores (see Ch. 6)
–za/2
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za/2
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Normal Dist. Critical Values
The value za/2 separates an area of a/2 in
the right tail of the z-dist.
The value –za/2 separates an area of a/2 in
the left tail of the z-dist.
The subscript a/2 is simply a reminder that the zscore separates an area of a/2 in the tail.
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