Download (1-a)100% confidence interval for m

7-1
Chapter Seven
Confidence
Intervals
這一章完全靠
背公式套之
7-2
McGraw-Hill/Irwin
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Confidence Intervals
7.1 Large Sample Confidence Intervals for a
Population Mean
7.2 Small Sample Confidence Intervals for
a Population Mean
7.3 Sample Size Determination
7.4 Confidence Intervals for a Population
Proportion
*7.5 A Comparison of Confidence Intervals
and Tolerance Intervals
7-3
7.1 Large Sample Confidence Intervals
for a Mean
If the sampled population is normal or the sample size is
large (n  30), a (1-)100% confidence interval for m is
x  z/2

n
(1-a)100%=confidence level
If  is unknown and the sample size is large (n  30),
estimate by the sample standard deviation s and a
(1-)100% confidence interval for m is
x  z/2
7-4
s
n
背公式套之
Example: Large Sample Confidence
Interval for Mean
Example 7.1: Gas Mileage Case
背公式套之
95% Confidence Interval
s
0.7992
x  z 0.025
 31.5531  1.960
n
49
 31.5531  0.2238  [31.33, 31.78]
99% Confidence Interval
x  z 0.005
7-5
s
0.7992
 31.5531  2.575
n
49
 31.5531  0.2940  [31.26, 31.85]
Three 95.44% Confidence Intervals for m
Gas Mileage Example
7-6
best coverage
The Effect of  on Confidence
Interval Width
7-7
7.2 Small Sample Confidence
Intervals for a Mean
t 分配: small sample is the key word
If the sampled population is normally distributed with
mean m, then a (1-)100% confidence interval for m is
x  t /2
s
n
t /2 is the t point giving a right-hand tail area of /2 under
the t curve having n – 1 degrees of freedom.
7-8
Example: Small Sample Confidence
Interval for Mean
Example 7.4: Debt-to-Equity Ratios
查表 t
x  t 0.025
s
0.1921
 1.3433  2.145
n
15
 1.3433  0.1064  [1.2369, 1.4497]
7-9
Effect of Degrees of Freedom on the
t-distribution
df makes difference
As the number of degrees of freedom increases, the spread
of the t distribution decreases and the t curve approaches
the standard normal curve.
7-10
7.3 Sample Size Determination
A sample size
 z /2 
n=

 B 
2
背公式套之,從信
賴區間而來,誤差
界線, 民調用之,
1068
will yield an estimate of x, precise to within B units of m,
with 100(1-) confidence.
7-11
7.4 Confidence Intervals for a Population
Proportion
If the sample size n is large [npˆ and n(1 - pˆ ) both  5] , then
a (1-)100% confidence interval for p is
p̂  z/2
pˆ (1 - pˆ )
n -1
n
背公式套之；Bernouli trial,
condition is above, based on
central limit theory
7-12
Example: Confidence Interval for a
Proportion
Example 7.8: Phe-Mycin side effects
35
p =
=.175
200
p̂  z/2
pˆ (1 - pˆ )
(0.175)(0.825)
 0.175  1.960
n -1
200 - 1
 0.175  0.053  [0.122, 0.228]
7-13
Determining Sample Size for Confidence
Interval for p
背公式套之
A sample size
 z/2 
n = p(1 - p)

 B 
2
will yield an estimate p̂ , precise to within B units of p,
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.
7-14
*7.5 A Comparison of Confidence Intervals
and Tolerance Intervals
7-15
Summary: Selecting an Appropriate
Confidence Interval
for a Population Mean
7-16
Confidence Intervals
Summary:
7.1 Large Sample Confidence Intervals
for a Population Mean
7.2 Small Sample Confidence Intervals
for a Population Mean
7.3 Sample Size Determination
7.4 Confidence Intervals for a Population
Proportion
*7.5 A Comparison of Confidence
Intervals and Tolerance Intervals
7-17