Monte Carlo Study

  Variable |     Obs        Mean   Std. Dev.       Min        Max
-----------+-------------------------------------------------------
Population |    1000    .5000000   .2886751   .0000000   1 .0000000
    n =  5 |    1000    .5036115   .125487    .106075      .8491938
    n = 20 |    1000    .5000545   .0637908   .2998779     .6954108
    n = 36 |    1000    .5044926   .048932    .3551762     .6496101

Here are plots of the population and each of the sampling distributions:

Figure 1: Population Distribution

Figure 2: Sampling Distribution for n = 5

Figure 3: Sampling Distribution for n = 20

Figure 4: Sampling Distribution for n = 36

Now let's try repeat the process of drawing 1,000 samples of n = 5, n = 20 and n = 36 from a highly skewed population:

  Variable |     Obs        Mean   Std. Dev.       Min        Max
-----------+-------------------------------------------------------
Population |    1000    1.088532   .6348374   .1219511   4.914284
    n =  5 |    1000    1.11774    .3052395   .3961306   2.660268
    n = 20 |    1000    1.116671   .1449324   .7067162   1.773374
    n = 36 |    1000    1.108628   .1043764   .8531538   1.515452

Here are plots of the population and each of the sampling distributions:

Figure 5: Population Distribution

Figure 6: Sampling Distribution for n = 5

Figure 7: Sampling Distribution for n = 20

Figure 8: Sampling Distribution for n = 36

Stata Example

use http://www.philender.com/courses/data/lahigh, clear

histogram daysatt, freq discrete



bs "summarize daysatt" "r(mean)",reps(1000) size(150) saving(mybs)

command:     summarize daysatt
statistic:   r(mean)
(obs=316)

Bootstrap statistics

Variable |   Reps   Observed       Bias   Std. Err.   [95% Conf. Interval]
---------+-------------------------------------------------------------------
     bs1 |   1000   74.65823  -.0141592   .9349639    72.82351  76.49294  (N)
         |                                            72.78667  76.37667  (P)
         |                                               72.72  76.34666 (BC)
-----------------------------------------------------------------------------
                              N = normal, P = percentile, BC = bias-corrected

use mybs

histogram bs1, freq

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