• 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
estimate ± margin of error
• 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 C confidence interval for the mean μ of a Normal population with known standard deviation σ, based on an SRS of size n, is given by
m=z*σ√n
Here z* is obtained from the row labeled z* at the bottom of Table D. The probability is C that a standard Normal random variable takes a value between −z* and z*. The confidence interval is
ˉx ± m
If the population is not Normal and n 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 C decreases,
• the sample size n increases, and
• the population standard deviation σ decreases.
• The sample size n required to obtain a confidence interval of specified margin of error m for a population mean is
n=(z*σm)2
where z* is the critical point for the desired level of confidence.
• 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.