第 8章 统计估计
(Statistical Estimation)
一、基本概念
1,Sample Statistic（样本统计量）
? Any number computed from sample data
? A random variable,Known
? Example,Average weekly food expenditures for 100
sampled residents
?Random? Yes! Due to randomness of sample selection
2,Population Parameter（总体参数）
? Any number computed for the entire population
? A fixed number,Unknown
? Example,mean weekly food expenditures for all 77,386
residents
?Do we ever know this? NO!
?But we estimate it (with error)
3,Estimator（估计量）
? A sample statistic used to guess a population
parameter
? Example,Sample average for 100 selected residents is an
estimator of the population mean of all 77,386 residents
Estimate [WRONG! Estimators are usually wrong,Often useful anyway]
? The actual number computed from the data
? Example,$33.91 is an estimate of neighborhood weekly
food expenditures per person
Estimation error:估计误差
? Estimator minus population parameter,Unknown
? Example,33.91 – 35.69 = –1.78
二,估计及其分布
1、计量资料
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nNX
nS
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n
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??
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2、计数资料
3、区间估计：参考值范围及可信区间（ RI,CI）
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nS
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n
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CISPSX
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SEPXSDPX
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%9958.2;58.2
%9596.1;96.1
%9064.1;64.1
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?????
?????
?????
???? ??
Mean,?,is
unknown
Population Random Sample I am 95%
confident that ? is
between 40 & 60.
Mean
X = 50
Estimation Process
估计过程
Sample
Estimate Population
Parameter...
with Sample
Statistic
Mean ?
Proportion p ps
Variance s2
Population Parameters Estimated
总体参数估计
?2
Difference ? - ?1 2 x - x 1 2
X
_
__
? Provides Range of Values
? Based on Observations from 1 Sample
? Gives Information about Closeness
to Unknown Population Parameter
? Stated in terms of Probability
Never 100% Sure
Confidence Interval Estimation
置信区间估计
Confidence Interval Sample Statistic
Confidence Limit
(Lower)
Confidence Limit
(Upper)
A Probability That the Population Parameter Falls
Somewhere Within the Interval.
Elements of Confidence Interval
Estimation（原理）
Parameter =
Statistic ± Its Error
Confidence Limits for Population Mean
总体均数的置信限
?? X? Error
= Error = X??
XX
XZ
??
? ???
xZ ??
XZX ?? ??
Error
Error
??X
90% Samples
95% Samples
?x_
Confidence Intervals
置信区间
xx,,?????? 64516451
xx ???? 96.196.1 ??
xx,,???? 582582 ??99% Samples
n
ZXZX X
?
? ?????
X
_
? Probability that the unknown
population parameter falls within the
interval
? Denoted (1 - ?) % = level of confidence
e.g,90%,95%,99%
? ????Is Probability That the Parameter Is Not Within
the Interval
Level of Confidence
置信水平
Confidence Intervals
Intervals
Extend from (1 - ?) % of
Intervals
Contain ?,
??% Do Not.
1 -?? ?/2?/2
X
_
?x_
Intervals & Level of Confidence
区间与置信水平
Sampling
Distribution of
the Mean
to
XZX ??
XZX ??
??? X
? Data Variation
measured by ?
? Sample Size
? Level of Confidence
(1 - ?)
Intervals Extend from
Factors Affecting Interval Width
影响置信区间宽度的因素
X - Z? to X + Z ? xx
n/XX ???
Mean
? Unknown
Confidence
Intervals
Proportion
Finite
Population? Known
Confidence Interval Estimates
置信区间估计
? Assumptions
? Population Standard Deviation Is Known
? Population Is Normally Distributed
? If Not Normal,use large samples
? Confidence Interval Estimate
Confidence Intervals (??Known)
总体方差已知时的置信区间
n
ZX / ??? ? 2???
n
ZX / ??? ? 2
? Assumptions
? Population Standard Deviation Is Unknown
? Population Must Be Normally Distributed
? Use Student抯 t Distribution
? Confidence Interval Estimate
Confidence Intervals (??Unknown)
总体方差未知时的可信区间
n
StX
n,/ ?? ?? 12?? ? n
StX
n,/ ?? ?? 12
Z
t0
t (df = 5)
Standard
Normal
t (df = 13)Bell-Shaped
Symmetric
Scatter?Tails
Student’s t Distribution
学生 t分布
? Number of Observations that Are Free
to Vary After Sample Mean Has Been
Calculated
? Example
? Mean of 3 Numbers Is 2
X1 = 1 (or Any Number)
X2 = 2 (or Any Number)
X3 = 3 (Cannot Vary)
Mean = 2
degrees of freedom = n
-1
= 3 -1
= 2
Degrees of Freedom (df)
自由度
Upper Tail Area
df,25,10,05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
3 0.765 1.638 2.353
t0
Assume,n = 3 df
= n - 1 = 2
?? =,10
??/2 =.05
2.920t Values
? / 2
.05
Student’s t Table
学生 t 分布表
A random sample of n = 25 has = 50 and
s = 8,Set up a 95% confidence interval
estimate for ?.
? ??.,46 69 53 30
X
Example,Interval Estimation
???Unknown
n
StX
n,/ ?? ?? 12 ??? n
StX
n,/ ?? ?? 12
25
80639250 ??,
??? 25
80639250 ??,
? Assumptions
? Sample Is Large Relative to Population
?n / N >,05
? Use Finite Population Correction Factor
? Confidence Interval (Mean,?X Unknown)
X? ??
Estimation for Finite Populations
有限总体的估计
n
StX
n,/ ?? ?? 12 n
StX
n,/ ?? ?? 121?
??
N
nN
1?
??
N
nN
? Assumptions
? Two Categorical Outcomes
? Population Follows Binomial Distribution
? Normal Approximation Can Be Used
? np??5 & n(1 - p) ? 5
? Confidence Interval Estimate
Confidence Interval Estimate Proportion
构成比的可信区间
n
)p(pZp ss
/s
???
?
1
2?? p n
)p(pZp ss
/s
???
?
1
2
A random sample of 400 Voters showed 32
preferred Candidate A,Set up a 95%
confidence interval estimate for p.
p? ?.053,107
Example,Estimating Proportion
n
)p(pZp ss
/s
???
?
1
2?? p n
)p(pZp ss
/s
???
?
1
2
4 0 0
0810896108 ).(..,???
400
0810896108 ).(..,???
?? p
样本含量何去何从？
Too Big:
?Requires too
much resources
Too Small:
?Won’t do
the job
What sample size is needed to be 90% confident
of being correct within ± 5? A pilot study
suggested that the standard deviation is 45.
n ZError? ? ? ?
2 2
2
2 2
2
1645 45
5
219 2 220?,,
Example,Sample Size for
Mean
Round Up
What sample size is needed to be within ± 5 with
90% confidence? Out of a population of 1,000,we
randomly selected 100 of which 30 were defective.
Example,Sample Size
for Proportion
Round Up
32 2 7
05
70306 4 511
2
2
2
2
.
.
)) (,(..
e r r or
)p(pZn ????
228?
What sample size is needed to be 90% confident
of being correct within ± 5? Suppose the
population size N = 500.
Example,Sample Size
for Mean Using fpc
Round Up
6152
15002219
5002219
10
0,
)(.
.
)N(n
Nnn ?
??
??
??
?
where 22192
22
0,er r o r
Zn ?? ?153?
本章小结
?Discussed Confidence Interval Estimation for
the Mean (??Known)
?Discussed Confidence Interval Estimation for
the Mean (??Unknown)
?Addressed Confidence Interval Estimation for
the Proportion
?Addressed the Situation of Finite Populations
?Determined Sample Size