# Chapter 1 Making Economic Decisions by HC12091303639

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```									        Chapter 14

Single-Population Estimation
Population Statistics
• Population Statistics:  , usually unknown
• Using Sample Statistics to estimate population
statistics:
– Point estimates       x
ˆ
x   
ˆ
R
 
ˆ
s
k         d2       c4
– Confidence interval
Confidence Interval
• Wrong Meaning:
Probability within the
confidence interval is
..%
• True meaning: The
interval will contain
the population
statistics ..% times
Confidence Interval for
Population Mean
Confidence Interval
• If  is known,
                
x  z       x  z1
2
n                2
n
• If  is unknown,
s                          s
x  t , n 1      x  t   2, n 1
2
n                           n
Common Confidence Levels
Confidence
1-        /2    1-/2                       z
Level
0.900   1.282
90%       .90   .10   .05     .95
0.950   1.645
95%       .95   .05   .025   .975               0.975   1.960
99%       .99   .01   .005   .995               0.990   2.326
0.995   2.576

0.100   -1.282
0.050   -1.645
1-                                    0.025   -1.960
/2                         /2                1-   0.010   -2.326
0.005   -2.576

-z               z                   -z
Example 14.2
=350, n=40, x  840
                   
x  z 2       x  z1 2
n                   n
• 90% CI
350                   350
840  1.645           840  1.645       748.97    931.03
40                    40

• 95% CI
350                   350
840  1.960          840  1.960       731.54    948.46
40                    40

• 99% CI
840  2.576
350
   840  2.576
350   697.45    982.55
40                    40
Confidence Interval for
Population Mean – Sample size
Can be (1- ) confident that the error | x   |
will not exceed a specified amount E when sample
size is

 z 2
2

n
 E


         

• Round up to an integer
Confidence Interval for
Population Mean
One-sided Confidence Bounds
• If  is known,
         
  x  z1    x  z   
n                   n
• If  is unknown,
s                        s
  x  t , n 1            x  t , n 1      
n                        n
t Distribution
• Let x1, x2, .. xn be a random sample from a normal distribution
with unknown  and 2, the random variable       x 
S n
has a t distribution with n-1 degrees of freedom.
• Probability Density Function, with k degrees of freedom,
[(k  1) 2]         1
f (x)                  2          ( k 1) / 2
  x  
k (k 2) [( x / k )  1]

• Mean      E( x )  0

• Variance V ( x )  k (k  2)
t Distribution

www.boost.org/.../graphs/students_t_pdf.png
t Distribution

df    90%     95%     99%
• Confidence Level: 1-
2     2.920   4.303   9.925
• Two-sided CI
5     2.015   2.571   4.032
10    1.812   2.228   3.169
30    1.697   2.042   2.750
50    1.676   2.009   2.678
100    1.660   1.984   2.626
1000   1.646   1.962   2.581
z     1.645   1.960   2.576
Example 14.4
x  52.34                                                   s                       s
x  t 2, n 1         x  t 2, n 1
• 95% CI, s=5.79, n=11                                       n                       n
5.79                       5.79
52 .34  2.2281           52 .34  2.2281               48.45    56.23
11                         11

• 95% CI, n=80                                                  Excel: tinv(, df)
5.79                       5.79
52 .34  1.9904           52 .34  1.9904                51.05    53.63
80                         80
• 95% CI, =5.79
5.79                      5.79
52 .34  1.960         52 .34  1.960                      48.92    55.76
11                        11
5.79                      5.79
52 .34  1.960         52 .34  1.960                       51.07    53.61
80                        80
Confidence Interval for
Population Variance
• Confidence Interval
(n  1)s 2         (n  1)s 2
2  2
 2 2, n 1       1 2, n 1
– Where: n-1 is the degree of freedom, and
1- is the confidence level
• One-sided Confidence Bounds
(n  1)s 2                           (n  1)s 2
   2
2 
   2
 , n 1                     21 , n 1
2 Distribution
• Let x1, x2, .. xn be a random sample from a normal distribution
with  and 2, and let s2 be the sample variance, then the
random variable (n-1)s2/2 has 2 distribution with n-1 degrees
of freedom.
• Probability Density Function, with k degrees of freedom,
1
f (x)     k
x ( k / 2)1e  x / 2   x 0
2 2 (k 2)
• Mean         E( x )  k

• Variance          V ( x )  2k

• Mode = k-2 (when k 3)
2 Distribution

Example 14.11
n=24, s2=.47
99% Confidence Interval
(n  1)s 2              (n  1)s 2
2 
 2 2, n 1            21 2, n 1
Excel: chiinv(, df)
 2.005 , 23  44 .1813        2.995 , 23  9.2604
(24  1)(.47)       (24  1)(.47)
2 
44.1813             9.2604

.2447   2  1.1673
Confidence Interval for
Population Proportion
Confidence Interval
p(1  p )
ˆ     ˆ                    p(1  p )
ˆ     ˆ
p  z 2
ˆ                      p  p  z1 2
ˆ
n                          n
One-sided Confidence Bound

p(1  p )
ˆ     ˆ                          p(1  p )
ˆ     ˆ
p  z
ˆ                     p      p  p  z1
ˆ
n                                n
Example 14.7
n=350, x=212
95% Confidence Interval
p  212 / 350  .6057
ˆ

p(1  p )
ˆ     ˆ                    p(1  p )
ˆ     ˆ
p  z 2
ˆ                     p  p  z1 2
ˆ
n                          n
.6057 (1  .6057 )                    .6057 (1  .6057 )
.6057  1.96                       p  .6057  1.96
350                                    350

.5545  p  .6569
Confidence Interval for
Population Proportion – Sample size
Can be (1- ) confident that the error | p  p |
ˆ
will not exceed a specified amount E when sample
size is
2
 z1 2 
n
 E 
 p(1  p )
        

• Round up to an integer
Confidence Interval for
Population Proportion – when p is small
Wilson Estimator
~ x2
p
n4
Confidence Interval
~     ~
p(1  p )                  ~     ~
p(1  p )
~
p  z 2                   ~
 p  p  z1 2
n4                        n4
Example 14.8
n=350, x=212
95% Confidence Interval with Wilson Estimator
~ 212  2  .6045
p
350  4
~     ~                    ~     ~
~  z p(1  p )  p  p  z
p 2                  ~          p(1  p )
1 2
n4                        n4
.6045 (1  .6045 )                    .6045 (1  .6045 )
.6045  1.96                       p  .6045  1.96
354                                    354

.5536  p  .6554
Example 14.9
N=50, x=1
99% Confidence Interval with Wilson Estimator
~ 1  2  .0556
p
50  4
~     ~                    ~     ~
~  z p(1  p )  p  p  z
p 2                  ~          p(1  p )
1 2
n4                        n4
.0556 (1  .0556 )                     .0556 (1  .0556 )
.0556  2.576                       p  .0556  2.576
54                                     54

 .0247  p  .1539            0  p  .1539

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