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What is a Confidence Interval?
Sampling Distribution of the Sample Mean
The statistic x estimates the population mean 
We want the sampling distribution to be centered at the
value of the parameter and to have little variation
Facts about sampling distribution of x
x  
x 

n
Notice that as n increases the sample to sample
variability in x decreases
If our sample comes from a normal distribution with
mean  and standard deviation  then
Z
x

has a standard normal distribution
n
Central Limit Theorem
If we sample from a population with mean  and
standard deviation  then
x   is approximately standard
Z
normal for large n

n
If n = 30 or larger, the central limit theorem will apply in
almost all cases
Example
A population of soft drink cans has amounts of liquid
following a normal distribution with   12 and
  0.2 oz.
What is the probability that a single can is between 11.9
and 12.1 oz.
What is the probability that x is between 11.9 and 12.1 for
n = 16 cans
Example
A population of trees have heights with a mean of 110
feet and a standard deviation of 20 feet
A sample of 100 trees is selected
Find  x
Find  x
Find P( x  108 feet)
What about P( X  108)
Sampling Distribution of the Sample Proportion
Population Proportion
# in population with characteristic
p
# in population
Sample Proportion
# in sample with characteristic
pˆ 
n
p̂ is a point estimate of p
 pˆ  p
 pˆ 
p1  p 
n
If we sample from a population with a proportion of p,
then
pˆ  p
is approximately standard
Z
p 1  p normal for large n


n
Example
Suppose the president’s approval rating is 56% and we
look at samples of size 100
Find  p̂
Find  p̂
Example
A survey of 120 registered voters yields 60 who plan to
vote for the republican candidate
p = proportion of all voters who plan to vote for the
republican candidate
Calculate the point estimate for p
Calculate the margin of error
Can we calculate the variance of the sampling
distribution
Do you see where the margin of error comes from?
Estimating Proportions with Confidence
The population proportion p is an unknown parameter
We wish to estimate p based on a sample
p̂ is a statistic which estimates p
We call p̂ a point estimate because its value is a point on
the real line
Unfortunately, for a continuous distribution the
ˆ  p is 0 because there is zero
probability that p
probability for any point
Statisticians prefer interval estimates
pˆ  E
E (error tolerance) depends on the sample size, how
certain we want to be, and the amount of variability in
the data
pˆ (1  pˆ )
EZ
n
The degree of certainty (probability that we are correct)
is known as the Level of Confidence
 (level of significance) is one minus the level of
confidence
Notice that increasing the level of confidence, decreases
the (level of significance) probability of being
incorrect and increases the width of the interval
All confidence intervals are two-sided probabilities with
a total area of 
Common Values for z
for 90% confidence z
for 95% confidence
2
 1.645
2
z  1.96
2
for 99% confidence z
 2.576
2
Example
A survey of 1,200 registered voters yields 540 who plan to
vote for the democratic candidate
Find a 95% confidence interval for p
We are 95% confident that the true proportion of voters
who will vote for the democratic candidate is between
42.2% and 47.8%