Transcript Confidence Intervals - McGraw Hill Higher Education

Chapter 8
Confidence Intervals
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
Confidence Intervals
8.1
8.2
8.3
8.4
8.5
8.6
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
Confidence Intervals for Parameters of Finite
Populations (Optional)
A Comparison of Confidence Intervals and
Tolerance Intervals (Optional)
8-2
LO 1: Calculate and
interpret a z-based
confidence interval for a
population mean when
is known.


8.1 z-Based Confidence
Intervals for a Mean: s 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
LO1
General Confidence Interval

In general, the probability is 1 – a that the population
mean m is contained in the interval
x  za 2 s x    x  za 2




s 

n
The normal point za/2 gives a right hand tail area under
the standard normal curve equal to a/2
The normal point - za/2 gives a left hand tail area under
the standard normal curve equal to a/2
The area under the standard normal curve between -za/2
and za/2 is 1 – a
8-4
LO1
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-a)100% confidence interval for m is
x  za

  x  za
n

s
2
s
2
n
, x  za
2
s 

n
8-5
LO 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 m
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
LO2
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
LO2
Degrees of Freedom and the
t-Distribution
As the number of degrees of freedom increases, the spread
of the t distribution decreases and the t curve approaches
the standard normal curve
8-8
LO2



Using the t Distribution
Table
Rows correspond to the different values of df
Columns correspond to different values of a
See Table 8.3, Tables A.4 and A.20 in Appendix A
and the table on the inside cover



Table 8.3 and A.4 gives t points for df 1 to 30, then for df =
40, 60, 120 and ∞
 On the row for ∞, the t points are the z points
Table A.20 gives t points for df from 1 to 100
 For df greater than 100, t points can be approximated by
the corresponding z points on the bottom row for df = ∞
Always look at the accompanying figure for guidance on
how to use the table
8-9
LO 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 (1a)100% confidence interval for m
is
x  ta

s
2
n
ta/2 is the t point giving a right-hand tail area of a/2
under the t curve having n1 degrees of freedom
8-10
LO 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
 za 2s 

n  

 B 
2
so that x is within B units of m, with 100(1a)% confidence
8-11
LO 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 a 2

p̂ 1  p̂ 
n
Here, n should be considered large if both


n · p̂ ≥ 5
n · (1 – p̂) ≥ 5
8-12
LO 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…
 za 2
n  p1  p 
 B




2
will yield an estimate p̂, precisely within B units of ρ,
with 100(1-a)% 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.
8-13
LO 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- a)100% confidence
interval for μ is
x  za 2

s
n
N n
N
A (1- a)100% confidence interval for the population
total is found by multiplying the lower and upper
limits of the corresponding interval for μ by N
8-14
LO 8: Distinguish
between confidence
intervals and tolerance
intervals (optional).

Tolerance interval: contains specified
percentage of individual population
measurements


8.6 A Comparison of Confidence
Intervals and Tolerance Intervals
Often 68.26%, 95.44%, 99.73%
Confidence interval: interval containing the
population mean μ, and the confidence level
expresses how sure we are that this interval
contains μ


Often level is set high (e.g., 95% or 99%)
Such a level is considered high enough to provide
convincing evidence about the value of μ
8-15