From Algorithms to Z-Scores - matloff - University of California, Davis

10.13. OTHER CONFIDENCE LEVELS 217
indicator variable ms signifying that the worker has a Master’s degree (but not a PhD). Let’s see
the difference between CS and EE on this variable:
> t . t e s t ( cs$ms , ee$ms )
Welch Two Sample t−t e s t
data : cs$ms and ee$ms
t = 2 . 4 8 9 5 , df = 1878.108 , p−value = 0.01288
a l t e r n a t i v e h y p o t h e s i s : true d i f f e r e n c e in means i s not equal to 0
95 percent c o n f i d e n c e i n t e r v a l :
0.01073580 0.09045689
sample e s t i m a t e s :
mean o f x mean o f y
0.3560551 0.3054588
So, in our sample, 35.6% and 30.5% of the two groups had Master’s degrees, and we are 95%
confident that the true population difference in proportions of Master’s degrees in the two groups
is between 0.01 and 0.09.
10.13 Other Confidence Levels
We have been using 95% as our confidence level. This is common, but of course not unique. We
can for instance use 90%, which gives us a narrower interval (in (10.22),we multiply by 1.65 instead
of by 1.96, which the reader should check), at the expense of lower confidence.
A confidence interval’s error rate is usually denoted by 1−α, so a 95% confidence level has α = 0.05.
10.14 One More Time: Why Do We Use Confidence Intervals?
After all the variations on a theme in the very long Section 10.2, it is easy to lose sight of the goal,
so let’s review:
Almost everyone is familiar with the term “margin of error,” given in every TV news report during
elections. The report will say something like, “In our poll, 62% stated that they plan to vote for Ms.
X. The margin of error is 3%.” Those two numbers, 62% and 3%, form the essence of confidence
intervals:
• The 62% figure is our estimate of p, the true population fraction of people who plan to vote
for Ms. X.