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%