what is a confidence interval? - chapter 21faculty.washington.edu/grover4/class14au12notes.pdf ·...
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What is a Confidence Interval?Chapter 21
November 19, 2012
Example
Confidence Interval
Example
1.0 Example
A political candidate wants to enter a primary in a district with100,000 eligible voters, but only if he has a good chance ofwinning. He hires a survey organization, which takes a S.R.S.of 2,500 voters. In the sample, 1,328 favor the candidate.
This gives a p̂ of
13282500× 100% = 53%.
Should he enter the primary?
1.1 Central Limit Theorem
The S.R.S. of size 2,500 would, if repeated many times,produce p̂ that closely follow the Normal density curve:
p
S.D.= √ [p(1-p) /2500]
The S.D. is small because the sample size is large. So almostall samples will produce a p̂ that is close to the true p.
1.2 68-95 Rule
By the 68-95 rule, 95% of all samples of size 2,500 give a p̂that lies in the interval
p − 2√
p(1−p)2500
, p + 2√
p(1−p)2500
p
p + 2 √ [ p(1-p)/2500 ]p - 2 √ [ p(1-p)/2500 ]
95% of all p-hat lie here
1.3 From p̂ to p
We can put this another way: 95% of all samples of size 2,500give a p̂ such that p is captured by the interval
p̂ − 2√
p(1−p)2500
, p̂ + 2√
p(1−p)2500
p
p - 2 √ [ p(1-p)/2500 ] p + 2 √ [ p(1-p)/2500 ]
2.0 Confidence Interval
When the population proportion has the value p, 95% ofall samples catch p in the interval extending 2 S.D.s oneither side of p̂.
That’s the interval
p̂ ± 2√
p(1−p)n
.
Unfortunately, this interval cannot be found only from ourdata. It depends on p!
But if n is large, then
p(1− p) ≈ p̂(1− p̂)
2.1 95% Confidence Interval
An approximate 95% confidence interval for p is:
p̂ ± 2√
p̂(1−p̂)n
.
It is only approximate because it holds only if n is largeenough.
Polling: Should the political candidate from example 1.0enter the primary?
2.2 Anatomy of a Confidence Interval
Our 95% confidence interval has the form:
estimate ± margin of error
Recall, we eye-balled the margin of error as 1/√n. Why?
M.O.E. = 2√
p(1−p)n
.
The largest value for p(1− p) occurs when p = 0.5. So
Largest M.O.E. = 1/√n.
2.3 Interpreting 95% Confidence
The first point to notice: confidence interval will dependon the particular sample drawn.
With some samples, the interval will cover the parameter.But with others it will fail to cover.
For about 95% of all samples, the interval
p̂ ± 2√
p̂(1−p̂)n
.
will cover the parameter p, for the other 5% it does not.
We are using a procedure that works 95% of the time.
2.3 Interpreting 95% Confidence
3.0 Example
Probabilities are used when reasoning from theto the .
Confidence levels are used when reasoning from theto the .
Choose from below to fill in the blanks in each of the twostatements above.
population sample
3.1 Example
A S.R.S. of 1,000 persons is taken to estimate the proportionof Democrats in a large population. It turns out that 543 ofthe people in the sample are Democrats. Calculate a 95%confidence interval for the proportion of Democrats in thepopulation.