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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.