SECTION 6.2 Summary

• The purpose of a confidence interval is to estimate an unknown parameter with an indication of how accurate the estimate is and of how confident we are that the result is correct. Any confidence interval has two parts: an interval computed from the data and a confidence level. The interval often has the form
  

• The confidence level states the probability that the method will give a correct answer. That is, if you use 95% confidence intervals, in the long run 95% of your intervals will contain the true parameter value. When you apply the method once (that is, to a single sample), you do not know if your interval gave a correct answer (this happens 95% of the time) or not (this happens 5% of the time).

• The margin of error for a level confidence interval for the mean of a Normal population with known standard deviation , based on an SRS of size , is given by

  Here, is obtained from the row labeled at the bottom of Table D. The probability is that a standard Normal random variable takes a value between and . The level confidence interval is

  If the population is not Normal and is large, the confidence level of this interval is approximately correct.

• Other things being equal, the margin of error of a confidence interval decreases as
  • - the confidence level decreases,
  • - the sample size increases, and
  • - the population standard deviation decreases.
• The sample size required to obtain a confidence interval of specified margin of error for a population mean is
  
  
  where is the critical point for the desired level of confidence.

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• A specific confidence interval formula is correct only under specific conditions. The most important conditions concern the method used to produce the data. Other factors such as the form of the population distribution may also be important. These conditions should be investigated prior to any calculations.