Chapter 1 Making Economic Decisions by HC12091303639

VIEWS: 0 PAGES: 22

									        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




fr.academic.ru/pictures/frwiki/67/Chi-square_..
                            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

								
To top