﻿The 
  sample 
  estimate 
  produces 
  the 
  true 
  population 
  mean 
  (X 
  = 
  x 
  = 
  11,25) 
  when 
  

   computed 
  in 
  this 
  manner. 
  The 
  bias 
  showoi 
  in 
  the 
  example 
  above 
  is 
  due 
  to 
  the 
  unequal 
  

   probability 
  of 
  each 
  member 
  of 
  the 
  population 
  to 
  be 
  selected 
  for 
  the 
  sample. 
  This 
  bias 
  

   also 
  influences 
  the 
  estimated 
  proportion 
  of 
  the 
  population 
  having 
  a 
  particular 
  attribute. 
  

   The 
  example 
  below 
  illustrates 
  this 
  point. 
  

  

  Example 
  2: 
  

  

  A 
  population 
  of 
  4, 
  500 
  skiers 
  has 
  the 
  characteristics 
  shown 
  below: 
  

  

  1,500 
  male 
  skiers 
  visiting 
  20 
  times 
  per 
  season 
  = 
  30,000 
  visits 
  

   500 
  female 
  skiers 
  visiting 
  20 
  times 
  per 
  season 
  = 
  10,000 
  visits 
  

   1,000 
  female 
  skiers 
  visiting 
  5 
  times 
  per 
  season 
  = 
  5,000 
  visits 
  

   1,500 
  male 
  skiers 
  visiting 
  10 
  times 
  per 
  season 
  = 
  15,000 
  visits 
  

  

  The 
  proportion 
  of 
  female 
  skiers 
  is: 
  

  

  P 
  = 
  1,500 
  / 
  4,500 
  = 
  33.3 
  percent 
  

  

  If 
  a 
  sample 
  of 
  6 
  percent 
  were 
  randomly 
  selected 
  on 
  a 
  day 
  when 
  400 
  were 
  skiing, 
  

   the 
  sample 
  would 
  probably 
  contain 
  24 
  skiers 
  with 
  the 
  following 
  characteristics: 
  

   12 
  male 
  skiers 
  who 
  each 
  visit 
  20 
  times 
  per 
  season 
  

   4 
  female 
  skiers 
  who 
  each 
  visit 
  20 
  times 
  per 
  season 
  

   2 
  female 
  skiers 
  who 
  each 
  visit 
  5 
  times 
  per 
  season 
  

   6 
  male 
  skiers 
  who 
  each 
  visit 
  10 
  times 
  per 
  season 
  

  

  Estimating 
  the 
  population 
  proportion 
  of 
  female 
  skiers 
  (P) 
  in 
  the 
  usual 
  manner 
  

   provides 
  the 
  sample 
  estimate: 
  P 
  = 
  6 
  / 
  24 
  = 
  25 
  percent. 
  Thus, 
  sample 
  estimates 
  of 
  

   population 
  proportions 
  or 
  attributes 
  are 
  also 
  affected 
  by 
  the 
  number 
  of 
  days 
  of 
  skiing 
  

   a 
  skier 
  does 
  during 
  a 
  season. 
  By 
  applying 
  the 
  method 
  suggested, 
  the 
  bias 
  is 
  removed. 
  

  

  (12 
  male 
  skiers) 
  (^) 
  =0.6 
  male 
  skiers 
  

  

  ( 
  4 
  female 
  skiers) 
  (^) 
  = 
  .2 
  female 
  skiers 
  

  

  ( 
  2 
  female 
  skiers) 
  (z-) 
  = 
  .4 
  female 
  skiers 
  

  

  ( 
  6 
  male 
  skiers) 
  (jq) 
  ~ 
  -6 
  male 
  skiers 
  

  

  Total 
  1.8 
  skiers 
  (0.6 
  female 
  skiers) 
  

  

  The 
  estimate, 
  therefore, 
  of 
  the 
  proportion 
  of 
  female 
  skiers 
  in 
  the 
  population 
  as 
  derived 
  

   from 
  the 
  sample 
  is 
  33.3 
  percent 
  (0.6 
  / 
  1.8). 
  

  

  The 
  sample 
  estimate 
  is 
  equal 
  to 
  the 
  population 
  parameter 
  (P 
  = 
  p 
  = 
  33.3 
  percent); 
  

   therefore, 
  it 
  is 
  unbiased. 
  This 
  weighting 
  method 
  was 
  used 
  to 
  adjust 
  estimates 
  of 
  skier 
  

   responses. 
  

  

  54 
  

  

  