Confidence Interval for a Mean
Confidence
Intervals
File Information: 25 Slides
_
_
X %C.I .  ( x  E    x  E )
Sample Size
 30
 30
Sigma
 known
  
x  z 

2
n


 unknown
x  t
  
x  z 

2
n


 s 


, df 
2
n


 s 
x  z 

2 n 
Confidence Interval for a Proportion

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
X %C.I .  ( p  E  p p  E )
  
 pq 
p  z 

2
 n 



Inferential Statistics:

INFERENTIAL STATISTICS: Uses sample data to
make estimates, decisions, predictions, or other
generalizations about the population.


The aim of inferential statistics is to make an inference
about a population, based on a sample (as opposed to a
census), AND to provide a measure of precision for the
method used to make the inference.
An inferential statement uses data from a sample and
applies it to a population.
Some Terminology


Estimation – is the process of estimating
the value of a parameter from information
obtained from a sample.
Estimators – sample measures (statistics)
that are used to estimate population
measures (parameters). A good estimator
should be:
unbiased
consistent
relatively efficient
Terminology (cont’d.)


Point Estimate – is a specific
numerical value estimate of a
parameter.
Interval Estimate – of a parameter is
an interval or range of values used to
estimate the parameter. It may or may
not contain the actual value of the
parameter being estimated.
Terminology (cont’d.)


Confidence Level – of an interval
estimate of a parameter is the probability
that the interval will contain the
parameter.
Confidence Interval – is a specific
interval estimate of a parameter
determined by using data obtained from a
sample and by using a specific confidence
level.
Situation #1: Large Samples or
Normally Distributed Small Samples




A population mean is unknown to us, and we wish to
estimate it.
Sample size is > 30, and the population standard
deviation is known or unknown.
OR sample size is < 30, the population standard
deviation is known, and the population is normally
distributed.
The sample is a simple random sample.
Confidence Interval for
(Situation #1)

A 1   confidence interval for
given by
  
x  z 

2 n 


is
Z/2 : Areas in the Tails
Obtaining :Convert
the Confidence Level
to a decimal, e.g. 95%
C.L. = .95. Then:
1    .95
  .05

  .025
2
2

95%
.025
.025

  
z 

2 n 
-z (here -1.96)
  
z 

2 n 
z (here 1.96)
2
Maximum Error of the Estimate

  
z
The term  2   is called the maximum error
 n
of estimate or margin of error. It is the
maximum likely difference between the point
estimate of a parameter and the actual value of
the parameter.
Consider

The mean paid attendance for a sample of
30 Major League All Star games was
$46,970.87, with a standard deviation of
$14,358.21. Find a 95% confidence
interval for the mean paid attendance at all
Major League All Star games.
95% Confidence Interval for the
Mean Paid Attendance at the Major
League All Star Games
 $14,358.21
$46,970.87  1.96

30


 $46,970.87  $5,138.02
($41,832.85    $52,108.89)
Minimum Sample Size Needed

For an interval estimate of the population mean
is given by
2
 z   
n 2 
 E 



Where E is the maximum error of estimate (margin
of error)
Situation #2: Small Samples




A population mean is unknown to us, and we wish to
estimate it.
Sample size is < 30, and the population standard
deviation is unknown.
The variable is normally or approximately normally
distributed.
The sample is a simple random sample.
Student t Distribution







Is bell-shaped.
Is symmetric about the mean.
The mean, median, and mode are equal to 0 and
are located at the center of the distribution.
Curve never touches the x-axis.
Variance is greater than 1.
As sample size increases, the t distribution
approaches the standard normal distribution.
Has n-1 degrees of freedom.
Student t Distributions for
n = 3 and n = 12
Student t
Standard
normal
distribution
distribution
with n = 12
Student t
distribution
with n = 3
0
Confidence Interval for
(Situation #2)

A 1   confidence interval for
given by
 s 
x  t  ,n1 

2
 n

is
Consider

The mean salary of a sample of n=12
commercial airline pilots is $97,334,
with a standard deviation of $17,747.
Find a 90% confidence interval for the
mean salary of all commercial airline
pilots.
90% Confidence Interval for the
Mean Salary of Commercial Airline Pilots
 $17,747 
$97,334  1.796

12 

 $97,334  $9,201.12
($88,132.88    $106,535.12)
t or z????
Is

Known?
yes
Use z-values no matter what
the sample size is.*
no
Is n greater than
or equal to 30?
yes
Use z-values and s in place
of  in the formula.
no
Use t-values and
s in the formula.**
*Variable must be normally distributed when n<30.
**Variable must be approximately normally distributed.
Situation #3: Confidence
Interval for a Proportion

Consider the following:
A USA Today Snapshots feature stated that
12% of the pleasure boats in the United
States were named Serenity.
Confidence Interval for a Proportion p

A confidence interval for a population proportion p, is
given by
pˆ  z 

Where
pˆ
2
pˆ qˆ
n
is the sample proportion .
qˆ  1  pˆ
n = sample size
np and nq must both be greater than
or equal to 5.
Consider


In a recent survey of 150 households, 54 had
central air conditioning. Find the 90% confidence
interval for the true proportion of households that
have central air conditioning.
Here
pˆ  54
 .36
150
qˆ  1  pˆ  1  .36  .64
n  150
(.36)(.64)
.36  1.645
150
 .36  .0645
(.296  p  .425)
We can be 90% confident that the true proportion, p, of
all homes having central air conditioning is between 29.6%
and 42.5%
Minimum Sample Size Needed

For an interval estimate of a population proportion
is given by
2
 z 
n  pˆ qˆ  2 
 E 


Where E is the maximum error of estimate (margin
of error)
Confidence and Precision



The length of a confidence interval is
the difference between the upper
bound and lower bound of the interval.
The maximum error of estimate
(margin of error) is equal to one half
the length of the confidence interval.
A shorter interval is a more precise
interval.