-- confidence intervals

Here  our goal is to find estimates  for the true value of  parameters  that  describe a  certain  population. Such  parameters  for  example  could  be  the  mean  m of all values of some quantitative population variable characteristic X, or its standard  deviation  s.  Or  it could be the population  proportion  p of individuals who share some qualitative variable characteristic.

The  following  two  examples  show  the  type  of  questions  we  are  attempting  to  give answers  to  here.

  1. What  is  the  average  weight  of  a  new born baby in the U.S?
  2. What  proportion  of  American  college  students  are regular smokers?
-- Point estimation

-- A  point  estimate  is  a  single  number  that  represents our  best guess for  the  value  of  the  parameter whose value we are trying to estimate.

The method for  finding a point-estimate for a population parameter involves the following steps:

  1. Draw  a random sample  from  the  population.
  2. Choose an appropriate sample statistic and compute its value from  the observed sample data.
  3. Use the value of the sample statistic to estimate the value of the population parameter in question.
-- point  estimators
A  point  estimator  is  a  sample  statistic  whose value  is  used  to  estimate  the  true value of a population  parameter.   A  list  of  the  most  common  estimators  used, is  shown  below.

 -- interval  estimation

-- An  interval  estimate  is  a  range of  values  which is constructed in such a way that it has a  high probability C,  called the confidence level, of  including  the  true value of the population  parameter whose value we are trying to estimate.

The method for constructing a confidence interval involves the following steps:

  1. Draw  a random sample from  the  population.
  2. Choose a statistic (usually this is one of the parameter-estimators we use to give point-estimates for the population parameter in question) and compute its value  from  the  observed  sample  data.
  3. The confidence interval  (range of values) for the parameter under investigation  is then given by:
estimate  ±  margin of error

--The  margin  of  error  depends  on  the  desired  level of confidence, and the  standarderror of  the estimate (i.e., the standard deviation of the distribution of the sample statistic). 


 -- confidence  interval  for  m ;  s  known

Assumptions:

  1. The  population  is  normally  distributed, or  the  sample-size n  is  larger than 30.
  2. The sample consists of  n randomly chosen and independent observations that are identically distributed.
Here z* is the upper (1 - C) / 2 critical  z-value  of the standard normal distribution -- the distribution of the standardized values of the sample mean. ( note: z* is also denoted by  za/2  where  a = 1 - C )

Sample  size  for  a  specified  margin  of  error   m  -  when  estimating m.

The  confidence  interval  for  the  population  mean  m  will have  a  specified  margin  of error  m   when  the  sample  size  is:


-- confidence  interval  for  m ;   s  not  known
Assumptions:
  1. The  population  is  normally  distributed.
  2. The sample consists of  n randomly chosen and independent observations that are identically distributed.
Here  t*  is  the upper (1 - C) / 2 critical t-value of the student t-distribution with n - 1 degrees of freedom -- the distribution of the standardized values of the sample mean, this time using the estimated standard deviation. For  the  most  common values  of  C,  values of  t* are  found  in  tables. ( note: t* is also denoted by  tn-1,a/2  where  a = 1 - C )

Note: For values of  n  larger than  30  the values of  t* are approximately equal to the corresponding values of  z*.  Also, with large values of  n  the normality assumption can be dropped, with the understanding that in this case the procedure results in an approximate level C confidence interval. 


-- confidence  intervals  for  p .

Note:  p   here  stands  for  the  true  proportion  (fraction, also expressed as %)  of  all members in  a population  of  a particular attribute. p is most commonly known as the "population proportion"

Note:  z*  is  again  the  z-value  associated  with  the  desired  level  of  confidence  C.
 

Sample  size  for  a  specified  margin  of  error   m  -  when  estimating  p.

The level  C  confidence  interval  for  the  population  proportion  p  will  have   margin  of  error at  most   m,  if  n is  chosen  to  be:


Nick Psomas, Fall 1998