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