Confidence Intervals for the Mean ( )

We assume that the standard deviation of the distribution is known, and compute a confidence interval using a point estimator for the mean , such as as follows:

1. Determine the significance level desired (this is the probability of missing the parameter value!). The confidence level is defined as .
2. Find : the position along the standard normal such that of the total area is found to the right of .
3. Create the confidence limits, left and right:

   

Note: The larger the sample size, the tighter the limits (the smaller the confidence interval).

We make use of this principle if we seek to find a confidence interval of fixed size. For example, suppose that you want to find a confidence interval of at most size 1 ounce about the quantity of beer in a cup. You may believe that the 18 ounce beer that's promised is actually no more than 17 ounces (and more than likely 16 ounces). So you seek to demonstrate this fact with a significance level of . We can solve to make the bounds around ,

equal W in size (in the beer example we'd set W to 1). To do this, solve

for n, from which we get

This tells us how to select our sample size so as to guarantee a given sized confidence interval.