Download Lecture 33 - Confidence Intervals Proportion

Confidence Interval
Estimation for a
Population Proportion
Lecture 33
Section 9.4
Tue, Mar 28, 2006
Point Estimates
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Point estimate – A single value of the statistic
used to estimate the parameter.
The problem with point estimates is that we
have no idea how close we can expect them to
be to the parameter.
That is, we have no idea of how large the error
may be.
Interval Estimates
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Interval estimate – an interval of numbers that
has a stated probability (often 95%) of
containing the parameter.
An interval estimate is more informative than a
point estimate.
Interval Estimates
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Confidence level – The probability that is
associated with the interval.
If the confidence level is 95%, then the interval
is called a 95% confidence interval.
Approximate 95% Confidence
Intervals
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How do we find a 95% confidence interval for p?
Begin with the sample size n and the sampling
distribution of p^.
We know that the sampling distribution is normal
with mean p and standard deviation
 pˆ 
p1  p 
n
The Target Analogy
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Suppose a shooter hits within 4 rings (4 inches)
of the bull’s eye 95% of the time.
Then each individual shot has a 95% chance of
hitting within 4 inches.
The Target Analogy
The Target Analogy
The Target Analogy
The Target Analogy
The Target Analogy
The Target Analogy
The Target Analogy
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Now suppose we are shown where the shot hit,
but we are not shown where the bull’s eye is.
What is the probability that the bull’s eye is
within 4 inches of that shot?
The Target Analogy
The Target Analogy
The Target Analogy
Where is the bull’s eye?
The Target Analogy
4 inches
The Target Analogy
4 inches
95% chance that the
bull’s eye is within
this circle.
The Confidence Interval
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In a similar way, 95% of the sample proportions
p^ should lie within 1.96 standard deviations
(p^) of the parameter p.
The Confidence Interval
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
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Therefore, if we compute a single p^, then we
expect that there is a 95% chance that it lies
within a distance 1.96p^ of p.
The Confidence Interval
The Confidence Interval
The Confidence Interval
p^
Where is p?
The Confidence Interval
1.96 p^
p^
The Confidence Interval
1.96 p^
p^
95% chance that p is
within this interval
Approximate 95% Confidence
Intervals
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Thus, the confidence interval is
pˆ  1.96 pˆ
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The trouble is, to know p^, we must know p.
(See the formula for p^.)
The best we can do is to use p^ in place of p to
estimate p^.
Approximate 95% Confidence
Intervals
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That is,
 pˆ 
pˆ 1  pˆ 
n
This is called the standard error of p^ and is
denoted SE(p^).
 Now the 95% confidence interval is
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pˆ  1.96  SE pˆ 
Example
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Example 9.6, p. 585 – Study: Chronic Fatigue
Common.
Rework the problem supposing that 350 out of
3066 people reported that they suffer from
chronic fatigue syndrome.
How should we interpret the confidence
interval?
Standard Confidence Levels
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The standard confidence levels are 90%, 95%,
99%, and 99.9%. (See p. 588 and Table III, p.
A-6.)
Confidence Level
z
90%
1.645
95%
1.960
99%
2.576
99.9%
3.291
The Confidence Interval
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The confidence interval is given by the formula
pˆ  z  SE pˆ 
where z
Is given by the previous chart, or
 Is found in the normal table, or
 Is obtained using the invNorm function on the TI-83.
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Confidence Level
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Rework Example 9.6, p. 585, by computing a
90% confidence interval.
 99% confidence interval.
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Which one is widest?
In which one do we have the most confidence?
Probability of Error
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We use the symbol  to represent the
probability that the confidence interval is in
error.
That is,  is the probability that p is not in the
confidence interval.
In a 95% confidence interval,  = 0.05.
Probability of Error
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Thus, the area in each tail is /2.
Confidence
 invNorm(/2)
Level
90%
0.10
-1.645
95%
0.05
-1.960
99%
0.01
-2.576
99.9%
0.001
-3.291
Think About It
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Think About It, p. 586.
Computing a confidence interval is a procedure
that contains one step whose outcome is left to
chance. (Which step?)
Thus, the confidence interval itself is a random
variable.
Interpretation
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See p. 587.
“If we repeated this procedure over and over,
yielding many 95% confidence intervals for p,
we would expect that approximately 95% of
these intervals would contain p and
approximately 5% would not.”
Interpretation
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Compare this to shooting at the target, where
the probability of hitting it is 95%.
“If we shoot at the target over and over, yielding
many bullet holes, we would expect that
approximately 95% of these bullet holes would
be in the target and approximately 5% would
not.”
Interpretation
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On the other hand, if we see that a particular
shot hit the target, then what are the chances
that it hit the target?
On the other hand, if we see that a particular
shot missed the target, then what are the
chances that it hit the target?
So, for a shot that has already been fired, what is
the probability that it hit the target?
Interpretation
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Therefore, if we are given a particular confidence
interval, it either does or does not contain p.
That is, the probability is either 0% or 100%,
but we do not know which.
Therefore, we should not talk about the
probability that it contains p.
It is the procedure that has a 95% of producing a
confidence interval that contains p.
Which Confidence Interval is Best?
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Which is better?
A wider confidence interval, or
 A narrower confidence interval.
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Which is better?
A low level of confidence, or
 A high level of confidence.
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Which is better?
A smaller sample, or
 A larger sample.
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Which Confidence Interval is Best?
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What do we mean by “better”?
Is it possible to increase the level of confidence
and make the confidence narrower at the same
time?
TI-83 – Confidence Intervals
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The TI-83 will compute a confidence interval
for a population proportion.
Press STAT.
Select TESTS.
Select 1-PropZInt.
TI-83 – Confidence Intervals
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A display appears requesting information.
Enter x, the numerator of the sample
proportion.
Enter n, the sample size.
Enter the confidence level, as a decimal.
Select Calculate and press ENTER.
TI-83 – Confidence Intervals
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A display appears with several items.
The title “1-PropZInt.”
 The confidence interval, in interval notation.
 The sample proportion p^.
 The sample size.
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How would you find the margin of error?
TI-83 – Confidence Intervals
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Rework Example 9.6, p. 585, using the TI-83.