What is a Confidence Interval?
Sampling Distribution of the Sample Mean
The statistic estimates the population mean
We want the sampling distribution to be centered at the value of the parameter and to have little variation
x
Facts about sampling distribution of
Notice that as n increases the sample to sample variability in decreases
x x
nx
x
If our sample comes from a normal distribution with mean and standard deviation then
has a standard normal distribution
Central Limit TheoremIf we sample from a population with mean
and standard deviation then is approximately standard normal for large n
n
xZ
n
xZ
If n = 30 or larger, the central limit theorem will apply in almost all cases
ExampleA population of soft drink cans has amounts of
liquid following a normal distribution with and
What is the probability that a single can is between 11.9 and 12.1 oz.
What is the probability that is between 11.9 and 12.1 for n = 16 cans
12oz. 2.0
x
ExampleA 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
FindFindFindWhat about
xx
feet) 108( xP)108( XP
Sampling Distribution of the Sample Proportion
Population Proportion
Sample Proportion
populationin #
sticcharacteri with populationin #p
np
sticcharacteri with samplein #ˆ
is a point estimate of p
If we sample from a population with a proportion of p, then
is approximately standard
normal for large n
p̂
pp ˆ n
ppp
1ˆ
n
pp
ppZ
1
ˆ
ExampleSuppose the president’s approval rating is 56%
and we look at samples of size 100
FindFind
p̂p̂
ExampleA 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 pCalculate the margin of errorCan we calculate the variance of the sampling
distributionDo 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
is a statistic which estimates pp̂
We call a point estimate because its value is a point on the real line
Unfortunately, for a continuous distribution the
probability that is 0 because there is zero probability for any point
Statisticians prefer interval estimates
p̂
pp ˆ
E (error tolerance) depends on the sample size, how certain we want to be, and the amount of variability in the data
Epˆ
n
ppZE
)ˆ1(ˆ
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
for 90% confidence
for 95% confidence
for 99% confidence
2z
645.12z
96.12z
576.22z
ExampleA 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%
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