Error Bounds

  Let

Notes:

• If the error distribution is symmetric about zero, we can obtain the error bounds by solving for b in
  
  
  and then setting b₁ = -b and b₂ = b. In this case, b may also be interpreted as a bound for the error magnitude. Thus, if b is a 95% bound, then we have the assurance that the estimator is within b of the parameter (or estimand) with probability 0.95.
• When the error distribution is normal with zero mean, the error bounds are
  
  where and
• As an example, suppose the sample mean is used as an estimator of the population mean. When based on a random sample from a normal distribution, error bounds for this point estimator are
  
  
  since in this case
  
  In practice, the population standard deviation is usually unknown and so we use
  
  
  as estimated error bounds for the sample mean where S is the usual sample standard deviation.
• Similar error bounds may also be easily obtained for the usual estimators of a population proportion, difference of two means, difference of two proportions. Essentially, all you need to do is determine the standard errors of the relevant estimators.