Download Statistical Analysis – Chapter 5 “Central Limit

Statistical Analysis – Chapter 8
“Confidence Intervals”
Roderick Graham
Fashion Institute of Technology
The Logic Behind Confidence Intervals
An Example
 What if we wanted to know the creativity of FIT students,
but as researchers, we are the first to take these
measurements. We are not testing hypotheses and we
have no population value to compare to our sample.
What do we do in this case?

In this situation, our sample mean, X ,that we calculate is
the “best guess” or “best estimate” of the population.
Our sample of FIT students will represent all FIT
students.
The Logic Behind Confidence Intervals

Although the sample mean that we calculate is a good
estimate, we know that our sample is affected by
randomness (Maybe there are a few very creative
students in our sample that skew the numbers).

Therefore, we want to construct a confidence interval
that will allow us to say, conceptually:
“although I cannot be sure that my calculated mean is the
true value of the population, I can be sure with 99%
confidence that the REAL population value is between ___
and ____.”
The Logic Behind Confidence Intervals


Before the election in 2008, most experts had a good idea
who would win each state.
This is because a point estimate was done on small sample of
likely voters. Let’s see an example:
http://www.americanresearchgroup.com/pres2008/NY08.html
What Equations Do We Use?
E  Zc s
n
E  tc s
n
The zc’s and tc’s are associated with levels of probability
on the normal curve table.
z.95 = a z-score of 1.96
t.95 = a t-score, for df of 5, of 2.57
s = standard deviation, and n = the number of cases, or respondents
T-Table for Confidence Intervals
Look for your
desired
confidence
interval
Find your df
(degrees of
freedom).
Remember, this is
n – 1.
Calculating Confidence Intervals
Step 1 – Choosing a Confidence Level
 This is only a matter of choosing a z-score or t-score
 These scores are symbolized as Zc or Tc
 For sample sizes with an N under 30, you would use a tscore
Z Scores for Common
Confidence Intervals
Z – Score (30 or more)

90%
1.65
95%
1.96
99%
2.58
Question: Why can I not construct a similar table for tscores?
Calculating Confidence Intervals
Step 2
 Calculate E using these equations…
E  Zc s

n
E  tc s
n
This E is called “the maximum error of estimate”
Calculating Confidence Intervals
Step 3
 Solve for the confidence interval using this equation:
X  E    X E
The x-bars are the means from the sample.
The E’s are the maximum error of estimate you calculated with the prior equation
You do not calculate µ, this is the population value you cannot know
Sample Problem – 8.6
8.6 (A)
8.6 (B and C)
Sample Problems – 8.4
Question 8.4
END