Chapter 8 - Confidence intervals. After mastering the material in this chapter, you will be able to: Calculate and interpret a z-based confidence interval for a population mean when σ is known, describe the properties of the t distribution and use a t table, calculate and interpret a t-based confidence interval for a population mean when σ is unknown,...
Confidence Intervals
8.1 zBased Confidence Intervals for a
Population Mean: σ Known
8.2 tBased 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)
82
LO8-1: Calculate and
interpret a z-based
confidence interval for a
population mean when
σ is known.
8.1 zBased 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
83
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
84
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
is
x z 2 x z 2 ,x z 2
n n n
85
LO8-2: Describe the
properties of the t
distribution and use a t
table. 8.2 tBased 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
x
t
s n
If the population is normal, then for any
sample size n, this sampling distribution is
called the t distribution
86
LO8-2
The t Distribution
The curve of the t distribution is similar to
that of the standard normal curve
Symmetrical and bellshaped
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
87
LO8-3: Calculate and
interpret a t-based
confidence interval for a
population mean when
σ is unknown.
tBased Confidence Intervals for a Mean:
σ Unknown
If the sampled population is normally distributed
with mean , then a (1 )100% confidence interval
for is
s
x t 2
n
t is the t point giving a righthand tail area of /2
/2
under the t curve having n1 degrees of freedom
Figure 8.10 88
LO8-4: Determine the
appropriate sample size
when estimating a
8.3 Sample Size Determination (z)
population mean.
If σ is known, then a sample of size
2
z 2
n
B
so that is within B units of , with 100(1 )
% confidence
89
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 (1
a)100% confidence interval for ρ is
ˆp 1 ˆp
ˆp z 2
n
Here, n should be considered large if both
◦n ∙ p̂ ≥ 5
◦n ∙ (1 – p̂) ≥ 5
810
LO8-6: Determine the
appropriate sample size
Determining Sample Size for
when estimating a
population proportion.
Confidence Interval for ρ
A sample size given by the formula…
2
z 2
n p1 p
B
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
811
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
s N n
x z 2
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
812
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