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Chapter 8
Confidence Intervals
McGraw-Hill/Irwin
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
Confidence Intervals
8.1 z-Based Confidence Intervals for a
Population Mean: σ Known
8.2 t-Based Confidence Intervals for a
Population Mean: σ Unknown
8.3 Sample Size Determination
8.4 Confidence Intervals for a Population
Proportion
8.5 Confidence Intervals for Parameters of
Finite Populations (Optional)
8-2
LO8-1: Calculate and
interpret a z-based
confidence interval for a
population mean when
σ is known.
8.1 z-Based Confidence Intervals for a
Mean: σ Known
Confidence interval for a population mean is
an interval constructed around the sample
mean so we are reasonable sure that it
contains the population mean
 Any confidence interval is based on a
confidence level

8-3
LO8-1
General Confidence Interval

In general, the probability is 1 – α that the
population mean μ is contained in the interval
x  z 2 x 




  x  z

2
 

n
The normal point zα/2 gives a right hand tail area
under the standard normal curve equal to α/2
The normal point -zα/2 gives a left hand tail area
under the standard normal curve equal to a/2
The area under the standard normal curve between
zα/2 and zα/2 is 1 – α
8-4
LO8-1
General Confidence Interval



Continued
If a population has standard deviation σ
(known),
and if the population is normal or if sample
size is large (n  30), then …
… a (1-)100% confidence interval for m
is
x  z 2


 
  x  z 2
, x  z 2

n 
n
n

8-5
LO8-2: Describe the
properties of the t
distribution and use a t
table.

8.2 t-Based Confidence Intervals for a
Mean: σ Unknown
If σ is unknown (which is usually the case),
we can construct a confidence interval for μ
based on the sampling distribution of
t
x m
s

n
If the population is normal, then for any
sample size n, this sampling distribution is
called the t distribution
8-6
LO8-2
The t Distribution




The curve of the t distribution is similar to
that of the standard normal curve
Symmetrical and bell-shaped
The t distribution is more spread out than the
standard normal distribution
The spread of the t is given by the number
of degrees of freedom
◦ Denoted by df
◦ For a sample of size n, there are one fewer
degrees of freedom, that is, df = n – 1
8-7
LO8-3: Calculate and
interpret a t-based
confidence interval for a
population mean when
σ is unknown.

t-Based Confidence Intervals for a Mean:
σ Unknown
If the sampled population is normally distributed
with mean m, then a (1)100% confidence interval
for m is
x  t
s
2
n
is the t point giving a right-hand tail area of /2
under the t curve having n1 degrees of freedom
 t/2
Figure 8.10
8-8
LO8-4: Determine the
appropriate sample size
when estimating a
population mean.
8.3 Sample Size Determination (z)
If σ is known, then a sample of size
 z 2 

n  

B


2
so that  is within B units of m, with 100(1)% confidence
8-9
LO8-5: Calculate and
interpret a large sample
confidence interval for a
population proportion.

8.4 Confidence Intervals for a Population
Proportion
If the sample size n is large, then a (1a)100%
confidence interval for ρ is
p̂  z  2

p̂1  p̂ 
n
Here, n should be considered large if both
◦ n · p̂ ≥ 5
◦ n · (1 – p̂) ≥ 5
8-10
LO8-6: Determine the
appropriate sample size
when estimating a
population proportion.

Determining Sample Size for
Confidence Interval for ρ
A sample size given by the formula…
 z 2
n  p1  p 
 B




2
will yield an estimate p̂, precisely within B units of
ρ, with 100(1-)% confidence

Note that the formula requires a preliminary
estimate of p
◦ The conservative value of p=0.5 is generally used when
there is no prior information on p
8-11
LO8-7: Find and
interpret confidence
intervals for parameters
of finite populations
(Optional).

8.5 Confidence Intervals for Parameters
of Finite Populations (Optional)
For a large (n ≥ 30) random sample of
measurements selected without replacement
from a population of size N, a (1- )100%
confidence interval for μ is
x  z 2
s
N n
N
n
 A (1- )100% confidence interval for the
population total is found by multiplying the
lower and upper limits of the corresponding
interval for μ by N
8-12