Transcript Ch8
Chapter 8
Confidence Intervals
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Outline
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 A Comparison of Confidence Intervals and
Tolerance Intervals (Optional)
8-2
8.1 z-Based Confidence Intervals for
a Mean: σ Known
If a population is normally distributed with
mean μ and standard deviation σ, then the
sampling distribution of x is normal with
mean μx = μ and standard deviation
x
n
Use a normal curve as a model of the
sampling distribution of the sample mean
Exactly, because the population is normal
Approximately, by the Central Limit Theorem for
large samples
8-3
The Empirical Rule
68.26% of all possible sample means
are within one standard deviation of the
population mean
95.44% of all possible observed values
of x are within two standard deviations
of the population mean
99.73% of all possible observed values
of x are within three standard
deviations of the population mean
8-4
Example 8.1 The Car Mileage Case
Assume a sample size (n) of 5
Assume the population of all individual car
mileages is normally distributed
Assume the standard deviation (σ) is 0.8
0.8
x
0.358
n
5
The probability that x with be within ±0.7
(2σx≈0.7) of µ is 0.9544
8-5
The Car Mileage Case
Continued
Assume the sample mean is 31.3
That gives us an interval of [31.3 ± 0.7]
= [30.6, 32.0]
The probability is 0.9544 that the
interval [x ± 2σ] contains the population
mean µ
8-6
Generalizing
In the example, we found the probability that
m is contained in an interval of integer
multiples of x
More usual to specify the (integer) probability
and find the corresponding number of x
The probability that the confidence interval
will not contain the population mean m is
denoted by
In the mileage example, = 0.0456
8-7
Generalizing
Continued
The probability that the confidence interval
will contain the population mean μ is denoted
by 1 -
1 – is referred to as the confidence coefficient
(1 – ) 100% is called the confidence level
Usual to use two decimal point probabilities
for 1 –
Here, focus on 1 – = 0.95 or 0.99
8-8
General Confidence Interval
In general, the probability is 1 – that the
population mean m is contained in the interval
x z 2 x x z 2
n
The normal point z/2 gives a right hand tail area
under the standard normal curve equal to /2
The normal point - z/2 gives a left hand tail area
under the standard normal curve equal to /2
The area under the standard normal curve
between -z/2 and z/2 is 1 –
8-9
Sampling Distribution Of All Possible
Sample Means
Figure 8.2
8-10
z-Based Confidence Intervals for a
Mean with σ Known
x z
x z
n
2
2
n
, x z
2
n
8-11
95% Confidence Interval
x z0.025 x x 1.96
x 1.96
Figure 8.3
n
, x 1.96
n
n
8-12
99% Confidence Interval
x z0.025 x x 2.575
x 2.575
Figure 8.4
n
, x 2.575
n
n
8-13
The Effect of a on Confidence
Interval Width
z/2 = z0.025 = 1.96
Figures 8.2 to 8.4
z/2 = z0.005 = 2.575
8-14
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
8-15
The t Distribution
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 (df)
For a sample of size n, there are one
fewer degrees of freedom, that is, df =
n–1
8-16
Degrees of Freedom and the
t-Distribution
Figure 8.6
8-17
The t Distribution and Degrees of
Freedom
As the sample size n increases, the degrees
of freedom also increases
As the degrees of freedom increase, the
spread of the t curve decreases
As the degrees of freedom increases
indefinitely, the t curve approaches the
standard normal curve
If n ≥ 30, so df = n – 1 ≥ 29, the t curve is very
similar to the standard normal curve
8-18
t and Right Hand Tail Areas
t is the point on the horizontal axis
under the t curve that gives a right
hand tail equal to
So the value of t in a particular
situation depends on the right hand tail
area and the number of degrees of
freedom
8-19
t and Right Hand Tail Areas
Figure 8.7
8-20
Using the t Distribution Table
Table 8.3 (top)
8-21
t-Based Confidence Intervals for a
Mean: σ Unknown
x t 2
Figure 8.10
s
n
8-22
8.3 Sample Size Determination
If σ is known, then a sample of size
z 2
n
B
2
so that x is within B units of m, with
100(1-)% confidence
8-23
Example (Car Mileage Case)
z 2
n
B
1.96.8
27.32
.3
2
2
8-24
Sample Size Determination (t)
If σ is unknown and is estimated from s, then
a sample of size
t 2 s
n
B
2
so that x is within B units of m, with 100(1)% confidence. The number of degrees of
freedom for the t/2 point is the size of the
preliminary sample minus 1
8-25
Example 8.6: The Car Mileage Case
t 2 s 2.776.7583
n
B
.3
2
2
2
49.24
8-26
8.4 Confidence Intervals for a
Population Proportion
If the sample size n is large, then a (1)100% confidence interval for ρ is
pˆ z 2
pˆ 1 pˆ
n
Here, n should be considered large if both
n · p̂ ≥ 5
n · (1 – p̂) ≥ 5
8-27
Example 8.8: The Cheese Spread
Case
pˆ z 2
pˆ 1 pˆ
.063.937
.063 1.96
n
1,000
.063 .0151
.0479,.0781
8-28
Determining Sample Size for
Confidence Interval for ρ
A sample size given by the formula…
z 2
n p1 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
8-29
8.5 A Comparison of Confidence
Intervals and Tolerance Intervals
(Optional)
A tolerance interval contains a specified
percentage of individual population
measurements
Often 68.26%, 95.44%, 99.73%
A confidence interval is an interval that
contains the population mean μ, and the
confidence level expresses how sure we are
that this interval contains μ
Confidence level set high (95% or 99%)
Such a level is considered high enough to provide
convincing evidence about the value of μ
8-30
Selecting an Appropriate Confidence
Interval for a Population Mean
Figure 8.19
8-31