Transcript Chapter 2 Describing Data: Graphs and Tables

Confidence Intervals
Inferences about Population
Means and Proportions
Lecture Objectives
You should be able to:
1. Define Key terms including Confidence Level,
Margin of Error, Standard Error of Means
2. Compute confidence intervals for population
means and proportions
3. Explain in your own words the connection
between the Central Limit Theorem and the
computation of confidence intervals.
Central Limit Theorem
Regardless of the population distribution, the
distribution of the sample means is
approximately normal for sufficiently large
sample sizes (n>=30), with
x  
and
x  
n
Applying CLT
If the sample means are normally distributed,

what proportion of them are within ± 1 Standard Error?

what proportion of them are within ± 2 Standard Errors?
If you take just one sample from a population, how likely
is it that its mean will be within 2 SEs of the population
mean?
How likely is it that the population mean is within 2 SEs
of your sample mean?
Confidence Intervals
The population mean is within 2 SEs of the sample mean,
95% of the time.
Thus ,
x

is in the range defined by:
2*SE, about 95% of the time.
(2 *SE) is also called the Margin of Error (MOE).
95% is called the confidence level.
The Standard Normal Distribution
Standardized Histogram of X Bar
Normal Distribution with Mean 0 and Standard Error of 1
500
99.7%
Frequency
400
95%
300
200
100
0
68%
-4
-3
-2
-1
0
1
X Bar - Number of SEs from the Mean
2
3
Confidence Interval for Mean
In general, the confidence interval for
x  z.

is given by

n
is the sample mean
z is the confidence factor. It is the number of
standard errors one has to go from the mean in
order to include a certain percent of observations. For
95% confidence the value is 1.96 (approximately
2.00).
x

is the standard error of the sample means.
n
Confidence Interval for Mean [2]
Since  is generally not known we substitute the sample
standard deviation, ‘s’. This changes the distribution of the
sample means from z (standard normal) to a t-distribution, a
close relative.
x
t.
s
n
The t value is slightly larger than the z for a given
confidence level, thereby increasing the margin of error.
That is the price of using s in place of 
Example – CI for Mean
A sample of 49 gas stations nationwide shows average
price of unleaded is $ 3.87 and a standard deviation of $
0.15 . Estimate the mean price of gas nationwide with
95% confidence.
In Excel, compute t with 5% error and (n-1), or 48 degrees of freedom
=tinv(0.05,48) = 2.010635, rounded to 2.01.
95% CI for the Mean is:
x t
s
n
=3.87 ± [2.01 * (0.15/√49)] = $ 3.87 ± 0.043
Thus,
$3.827 <

< $3.913
Interpret the result!
CI for Proportions
For proportions,
p = population proportion
pˆ = sample proportion
Confidence Interval for p is given by
pˆ
±z.
pˆ (1  pˆ )
n
Note similarity to formula for Means.
What is the SE here?
Example - Proportions
The Wall Street Journal for Sept 10, 2008 reports that a
poll of 860 people shows a 46% support for Sen. Obama
as President.
Find the 95% CI for the proportion of the population
that supports him.
95% CI is:
0.46(1  0.46)
0.46  1.96
860
= 0.46 ± 0.033
Thus, .427 < p < .493
How big a sample?
A TV station hires you to conduct an opinion poll.
They want to know the proportion of the U.S.
population that believes that alien landings on
Earth have occurred.
You are to ensure that the margin of error in the
inference from the sample is as close to 3% as
possible, for a 95% confidence interval.
How big a sample do you need?
[sample size.xls].