### Ch8

```Chapter 8
Confidence Intervals
McGraw-Hill/Irwin
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
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  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
8-29
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
```