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Statistics for Business
(ENV)
Chapter 8
Confidence Intervals
1
Confidence Intervals
8.1
8.2
8.3
8.4
z-Based Confidence Intervals for a
Population Mean: σ Known
t-Based Confidence Intervals for a
Population Mean: σ Unknown
Sample Size Determination
Confidence Intervals for a Population
Proportion
Reminder: Sampling distribution
– If a population is normally distributed with mean
m and standard deviation σ, then the sampling
distribution of is normal with mean mM = m and
standard deviation
M  n
– Use a normal curve as a model of the sampling
distribution of the sample mean
• Exactly, if the population is normal
• Approximately, by the Central Limit Theorem
for large samples
Example: SAT scores
• The population of scores (X) on the SAT forms a
normal distribution with mean m= 500 and =
100.
• In a random sample of n = 25 students,
– the distribution of the sample mean is a normal
distribution with a mean=500
– and standard deviation is     100  20
x
n
25
4
In a random sampling of students of sample size=25,
we are confident that 80% of the samples will have a
mean between m - 1.28 x and m + 1.28  x , or
within the interval [474.4, 525.6]
5
In other words:
The probability that will be within ±25.6 of µ is
80%,
OR
If we know
then there is 80% probability that µ
will be within ±25.6 away from
OR
we are 80% confident that µ will be within an
interval ±25.6 away from
6
Point and Interval Estimates
A confidence interval is a range of values in which the
population parameter (say m) is expected to be there.
Usually, people consider the 95% and 99% confidence
intervals.
m is between ? and ?
A point estimate is a single value (sample statistic)
used to estimate a population parameter.
m
s
Example 3
The Dean of the Business School
wants to estimate the mean number
of hours students studied per week.
A sample of 49 students showed a
mean of 24 hours with a sd of 4
hours. What is the population mean?
The value of the population mean is unknown. Our best
estimate of this value is the sample mean of 24.0 hours. This
value is called a point estimate.


X m
P 0 
 1.96   47.5%
/ n




X m
P  1.96 
 1.96   95%
/ n


  1.96
1.96 
P
 X m 
  95%
n
n 


1.96
1.96 
P X 
mX
  95%
n
n 


1.96(4)
1.96(4) 
P 2.4 
 m  2.4 
  95%
49
49 


1.96(4)
1.96(4) 
P 2.4 
 m  2.4 
  95%
49
49 

P (22.88  m  25.12)  95%
Constructing General Confidence
Intervals for µ
X  z

n
95% CI for the (population) mean
X  1.96

n
X  2.58

n
99% CI for the (population) mean
95 percent confidence interval for
the population mean
X  1.96

 24.00  1.96
n
4
49
 24.00  1.12
The 95% CI for the mean is from 22.88 to 25.12.
What if we don’t know ?
However, normally, we don’t know the
population (sd) .
So, normally, people just replace
(estimate) the  by s.
It doesn't matter too much since normally
we are considering a large sample(n30).
Factors that determine the width of a
confidence interval
The sample size, n
The level of confidence, 1-
The s.d. of the population,  (usually
estimated by the sample s.d., or s)
Constructing General Confidence
Intervals for µ
Confidence interval for the mean (n < 30 and
the underlying distribution is normal)
X t
s
n
The value of t depends on the confidence level as
well as the degrees of freedom (df=n-1).
Characteristics of the t distribution
It is a continuous, bell-shaped and symmetrical distribution,
which is flatter than a normal distribution.
There is a family of t distributions, determined by its
degrees of freedom (n-1).
The t-distribution approaches N(0, 1) as n
approaches infinity.
Distributions of the t statistic for different values of
degrees of freedom are compared to a normal
distribution.
16
Confidence Interval for a Population
Proportion
Let X ~ Bin(N, ), then P =X/N is called a population
proportion.
The distribution for a population proportion. Both n
and n(1- ) > 5
  (1   ) 
P ~ N  ,

A point estimate of the population
proportion is given by the sample
proportion P .
Confidence Interval for a Population
Proportion, obtained from the
sample proportion P
N


P(1  P)
Pz
n
EXAMPLE 4
A sample of 500 executives who
own their own home revealed 175
planned to sell their house after
they retire. Develop a 98% CI for
the proportion of executives that
plan to sell their house.
Here, the sample proportion is p=175/500=0.35
(.35)(.65)
0.35  2.326
 .35  .0497
500
Selecting a Sample Size
Let E be the error term appear in the CI
Ez

n
E is also known as the width of the C.I
divided by 2.
Selecting a Sample Size
 z 
n

 E 
2
2
where
n is the size of the sample
E is the allowable error
z the z- value corresponding to the selected level of
confidence
 the population s.d..
Example 6
A consumer group would like to
estimate the mean monthly
electricity charge for a single
family house in July within $5
using a 99 percent level of
confidence. Based on similar
studies the s.d. is estimated to
be $20.00. How large a sample is
required?
2
 (2.58)( 20) 
n
  107
5


Sample Size for Proportions
The formula for determining the sample size in the
case of a proportion is
 Z
n  p(1  p) 
 E
where
p is the estimated proportion, based on past experience
or a pilot survey
z is the z value associated with the degree of confidence
selected
E is the maximum allowable error the researcher will
tolerate
2
Example 7
The American Kennel Club wanted to estimate the proportion of
children that have a dog as a pet. If the club wanted the estimate
to be within 3% of the population proportion, how many children
would they need to contact? Assume a 95% level of confidence
and that the club estimated that 30% of the children have a dog
as a pet.
2
 1.96 
n  (.30)(.70)
  897
 .03 
“The nationwide telephone survey was
conducted Friday through Wednesday with 1,224
adults and has a margin of sampling error of plus
or minus three percentage points.”
Nation’s Mood at Lowest Level in Two Years, Poll Shows
By JIM RUTENBERG and MEGAN THEE-BRENAN
Published: April 21, 2011
24
Chapter 8
Estimation and Confidence Intervals
When you have completed this chapter, you will be able
to:
ONE
Define what is meant by a point estimate.
TWO
Construct a confidence interval for the mean when the population
standard deviation  is known and the sample size is large enough
or underlying distribution is normal.
THREE
Construct a confidence interval for the mean when the population
standard deviation  is unknown and sample size is large enough or
underlying distribution is normal.
Chapter 8
continued
FOUR
Construct a confidence interval for the population proportion.
FIVE
Construct a confidence interval for the mean when the
population size is finite.
SIX
Determine the sample size for attribute and variable sampling.