Download 6.3 Day 1

Chapter 6
Confidence Intervals
§ 6.3
Confidence Intervals
for Population
Proportions
Point Estimate for Population p
The probability of success in a single trial of a binomial
experiment is p. This probability is a population
proportion.
The point estimate for p, the population proportion of
successes, is given by the proportion of successes in a
sample and is denoted by
pˆ  x
n
where x is the number of successes in the sample and n is
the number in the sample. The point estimate for the
proportion of failures is qˆ = 1 – p̂. The symbols p̂ and qˆ
are read as “p hat” and “q hat.”
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Point Estimate for Population p
Example:
In a survey of 1250 US adults, 450 of them said that their favorite
sport to watch is baseball. Find a point estimate for the population
proportion of US adults who say their favorite sport to watch is
baseball.
n = 1250
x = 450
pˆ  x  450  0.36
n 1250
The point estimate for the proportion of US adults who
say baseball is their favorite sport to watch is 0.36, or
36%.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Confidence Intervals for p
A c-confidence interval for the population proportion p is
pˆ  E  p  pˆ  E
where
E  zc
pq
ˆ ˆ.
n
The probability that the confidence interval contains p is c.
Example:
Construct a 90% confidence interval for the proportion of
US adults who say baseball is their favorite sport to watch.
n = 1250
x = 450
p̂  0.36
Continued.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Confidence Intervals for p
Example continued:
n = 1250
p̂  0.36
x = 450
ˆˆ
E  z c pq
n
qˆ  0.64
Left endpoint = ?
•
p̂  E  0.36  0.022
 0.338
 1.645
(0.36)(0.64)  0.022
1250
Right endpoint = ?
p̂  •
0.36
•
p̂  E  0.36  0.022
 0.382
With 90% confidence we can say that the proportion of all
US adults who say baseball is their favorite sport to watch
is between 33.8% and 38.2%.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Finding Confidence Intervals for p
Constructing a Confidence Interval for a Population Proportion
In Words
In Symbols
1. Identify the sample statistics n and x.
2. Find the point estimate p̂.
3. Verify that the sampling distribution
can be approximated by the normal
distribution.
4. Find the critical value zc that
corresponds to the given level of
confidence.
5. Find the margin of error E.
6. Find the left and right endpoints and
form the confidence interval.
pˆ  x
n
npˆ  5, nqˆ  5
Use the Standard
Normal Table.
ˆˆ
E  z c pq
n
Left endpoint: p̂  E
Right endpoint: p̂  E
Interval: pˆ  E  p  pˆ  E
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Sample Size
Given a c-confidence level and a margin of error, E, the
minimum sample size n, needed to estimate p is
2
 zc 
ˆˆ  .
n  pq
E
This formula assumes you have an estimate for p̂ and qˆ.
If not, use pˆ  0.5 and qˆ  0.5.
Example:
You wish to find out, with 95% confidence and within 2% of
the true population, the proportion of US adults who say
that baseball is their favorite sport to watch.
Continued.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Sample Size
Example continued:
You wish to find out, with 95% confidence and within 2% of
the true population, the proportion of US adults who say
that baseball is their favorite sport to watch.
n = 1250
x = 450
2
p̂  0.36
1.96 
z 
ˆ ˆ  c   (0.36)(0.64) 
n  pq

 0.02 
E
2
 2212.8 (Always round up.)
You should sample at least 2213 adults to be 95% confident.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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