Confidence Interval Test
Notation
We can say that
a) we are 95% confident that our x lies within a certain interval, or
b) we can say that our confidence interval has a level of significance of 5% implying
  .05 . Both a and b mean the same thing.
Example: A confidence interval with a level of significance of   0.2 means that
we are (1   ) 100% = 80% confident that our x lies within a certain interval
A significance level of   .05 means that our final conclusion has a 5% chance of being
incorrect. The 0.05 pertains to the total area in the two tails of our sampling distribution,
thus  / 2  .025 refers to the area in one tail. It is customary to notate the critical Z
value in terms of  / 2 , and we call this critical value Z / 2 .
The critical value Z / 2 is the number of standard deviations from  x   to
the border of  /2.
The error is E = ( Z / 2 ) (  x ) and the confidence interval about  is x  E or
i
x  E    x  E . This means that P( x  E    x  E ) = (1- )
Examples
Example 1: You wish to estimate the average size of a machine bolt. If your sample
statistics are n = 50, x = 2.2 cm, find a confidence interval for  at the  = 0.07 level of
significance. The population standard deviation is known to 1.3 cm.
Example 2: You wish to estimate the time it takes to complete a job. If your sample
statistics are n = 37, x = 27 hrs , s = 3.5 hrs., find an 87% confidence interval about  .
Example 3: (Proportion problem) You wish to estimate the proportion of females at USA.
If your sample statistics are n = 90, x = 60 females, find a confidence interval for p ( the
proportion of females in the population) at the  = 0.10 level of significance.