Transcript Point Estimator

Ch 10 實習
Concepts of Estimation


2
The objective of estimation is to determine
the value of a population parameter on the
basis of a sample statistic.
There are two types of estimators:
Point Estimator
Interval estimator
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Point Estimator
A point estimator draws inference about a
population by estimating the value of an
unknown parameter using a single value
or point.
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Point Estimator
A point estimator draws inference about a
population by estimating the value of an
unknown parameter using a single value
or point.
Parameter
Population distribution
?
Sampling distribution
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Point estimator
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Interval Estimator
An interval estimator draws inferences
about a population by estimating the value
of an unknown parameter using an
interval.
Population distribution
Parameter
Sample distribution
Interval estimator
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Estimator’s Characteristics

Selecting the right sample statistic to estimate a
parameter value depends on the characteristics of
the statistic.
Estimator’s desirable characteristics:
Unbiasedness: An unbiased estimator is one whose
expected value is equal to the parameter it estimates.
Consistency: An unbiased estimator is said to be
consistent if the difference between the estimator and
the parameter grows smaller as the sample size
increases.
Relative efficiency: For two unbiased estimators, the one
with a smaller variance is said to be relatively efficient.
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Unbiased Estimate
估計的量，我們會希望它不要系統性的高估、或者系統
性的低估，也就是要求估計量有「不偏」性質。用秤體重來
比喻的話，如果體重計有時把我們秤重了些、有時又秤輕了
些，但是若秤了許許多多次之後，平均起來就等於我們的真
實體重的話，就相當於有不偏性質

E (X )  
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E ( pˆ )  p
2
(x  x)
2
s 
n 1
E (s 2 )   2
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Consistency…
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
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An unbiased estimator is said to be
consistent if the difference between the
estimator and the parameter grows smaller
as the sample size grows larger.
E.g. X is a consistent estimator of
because: V(X) is
That is, as n grows larger, the variance of
X grows smaller.
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Relative Efficiency…
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
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If there are two unbiased estimators of a
parameter, the one whose variance is
smaller is said to be relatively efficient.
E.g. both the the sample median and
sample mean are unbiased estimators of
the population mean, however, the sample
median has a greater variance than the
sample mean, so we choose
since it is
relatively efficient when compared to the
sample median.
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Example 1

a.
b.
c.
d.
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A point estimate is defined as:
The average of the sample values
The average of the population values
A single value that is the best estimate of an
unknown population parameter
A single value that is the best estimate of an
unknown sample statistic
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Example 2

a.
b.
c.
d.
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Which of the following statements is correct?
The sample mean is an unbiased estimator
of the population mean
The sample proportion is an unbiased
estimator of the population proportion
The difference between two sample means
is an unbiased estimator of the difference
between two population means
All of the above
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Estimating the Population Mean when
the Population Variance is Known

How is an interval estimator produced from a
sampling distribution?
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A sample of size n is drawn from the population,
and its mean x is calculated.
By the central limit theorem x is normally
distributed (or approximately normally distributed.),
thus…
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Estimating the Population Mean when
the Population Variance is Known
x 
Z
 n
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We have established before that
P(  z  2
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

 x    z 2
)  1 
n
n
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The Confidence Interval for  (  is known)

This leads to the following
equivalent statement
P( x  z  2


   x  z 2
)  1 
n
n
The confidence interval
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Interpreting the Confidence Interval for 
1 –  of all the values of x obtained in repeated
sampling from a given distribution, construct an interval

 

x  z 2 n , x  z 2 n 


that includes (covers) the expected value of the
population.
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Graphical Demonstration of the Confidence
Interval for 
Confidence level
1-
x  z 2
Lower confidence limit
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
n
x
2z  2
x  z 2

n

n
Upper confidence limit
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The Confidence Interval for  (  is known)
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Four commonly used confidence levels
Confidence
level
0.90
0.95
0.98
0.99
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
0.10
0.05
0.02
0.01
/2
0.05
0.025
0.01
0.005
z/2
1.645
1.96
2.33
2.575
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The Width of the Confidence Interval
2z  2

n
The width of the confidence interval is
affected by
•
•
•
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The population standard deviation ()
The confidence level (1-)
The sample size (n).
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Example 3

a.
b.
c.
d.
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In developing an interval estimate for a
population mean, the population standard
deviation σ was assumed to be 10. The
interval estimate was 50.92±2.14. Had σ
equaled 20, the interval estimate would be
60.92±2.14
50.92±12.14
101.84±4.28
50.92±4.28
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Example 4
a. The mean of a random sample of 25 observations
from a normal population whose standard deviation
is 40 is 200. Estimate the population mean with
95% confidence.
b. Repeat part a changing the population standard
deviation to 25.
c. Repeat part a changing the population standard
deviation to 10
d. Describe what happens to the confidence interval
estimate when the standard deviation is decreased
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Solution

40
a. x  z / 2
 200  1.96(
)  200  15.68, LCL=184.32, UCL=215.68
n
25

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b. x  z / 2
 200  1.96(
)  200  9.8, LCL=190.2, UCL=209.8
n
25

10
c. x  z / 2
 200  1.96(
)  200  3.92, LCL=196.08, UCL=203.92
n
25
d . The interval narrows
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Example 5
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
The following data represent a random
sample of 9 marks (out of 10) on a statistics
quiz. The marks are normally distributed with
a standard deviation of 2. Estimate the
population mean with 90% confidence.
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4 6 8 5 4 8 3 10 9
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Solution
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
2
 6.33  1.645( )  6.33  1.10
n
9
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x  z / 2
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LCL=5.23, UCL=7.43
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Selecting the Sample size
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We can control the width of the confidence interval
by changing the sample size.
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Thus, we determine the interval width first, and
derive the required sample size.
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The phrase “estimate the mean to within W units”,
translates to an interval estimate of the form
xw
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Selecting the Sample size
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The required sample size to estimate the mean is
 z  2 
n

 w 
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2
Any non integer value must be rounded up.
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Example 6
a. A statistics practitioner would like to estimate
a population mean to within 10 units. The
confidence level has been set at 95% and 
=200. Determine the sample size
b. Suppose that the sample mean was
calculated as 500. Estimate the population
mean with 95% confidence
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Solution
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a. n=(zα/2×σ/W)2=(1.96×200/10)2=1537
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b. 500±10
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Example 7
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A statistics professor wants to compare today’s
students with those 25 years ago. All of his current
students’ marks are stored on a computer so that he
can easily determine the population mean. However,
the marks 25 years ago reside only in his musty files.
He does not want to retrieve all the marks and will
be satisfied with a 95% confidence interval estimate
of the mean mark 25 years ago. If he assumes that
the population standard deviation is 10, how large a
sample should he take to estimate the mean to
within 3 marks?
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Solution
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n=(zα/2×σ/W)2=(1.96×10/3)2=43
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