Transcript 5-1 Random Variables and Probability Distributions

8-2 Estimation
Estimating μ when σ is
UNKNOWN
Imagine
You are in charge of quality control at Guinness
Brewery in Dublin, Ireland. Your job is to make sure
that the stout is of high enough quality to meet the
demands of your customers.
You need to test the samples without losing too much
product…
But what if you are taking small samples the test results
are not quite right? Rejecting perfectly acceptable
batches?
PS NO you may not drink your samples… you are a
chemist!
William S. Gossett
This was his job.
He evaluated the quality of the stout based on
differences in the process (varieties of barley and
hops, drying methods, etc)
He was rejecting about 15% of good batches, which
was too high.
He knew s but not σ. He needed another method to
evaluate error.
He worked with a very famous statistician, Karl
Pearson, on understanding Standard Errors, and
developed the Student’s T distribution*
Student’s t distribution?
Guinness had been burned before by another
employee who had published trade secrets.
Therefore they instituted a policy that no
employee could publish results.
He used a pseudonym – Student.
Not sure why.
Is this different from z?
Yes and no.
There is a t-table in the back of your book.
Your calculator can calculate the t value and
the associated probability.
Find z when you know σ. (not likely)
Find t when you don’t. (more likely)
The process
Assuming x has a normal distribution with
mean μ. For sample size n with mean
 and standard deviation s, the t
variable is found by
The process
Assuming x has a normal distribution with
mean μ. For sample size n with mean
 and standard deviation s, the t
variable is found by
x μ
t
s
n
With a new parameter, called degrees
of freedom (d.f.) which = n – 1.
Compares to z?
Symmetric about μ = 0.
Bell Shaped.
Difference? The tails are a bit higher
You will be cross correlating with two things – the t
score itself AND the degrees of freedom.
NOTE: More degrees of freedom (i.e. More n) creates
a curve the resembles the standard normal
distribution. Your book has a good picture.
Also: the Empirical rule does not work for t models
that have a low number of degrees of freedom.
How to read
The top row is the c value
The side row is the degrees of freedom
(n – 1)
Certain standard c values are in a table.
Don’t worry about the indication “one tail”
and “two tail”. We’ll deal with that later.
And?
Just like yesterday,
x E  μ  x E
And?
Just like yesterday,
x E  μ  x E
And E  tc  σ
n
Remember to use n – 1 for d.f.
Example
How many calories are there in 3 ounces of
french fries? It depends on where you get
them. Good Cholesterol Bad Cholesterol by
Roth and Streicher gives the data from eight
popular fast-food restaurants. The data are
222
255
254
230
249
222
237
287
Use the data to create a 99% confidence
interval for the mean calorie count in 3
ounces of fries.
I love fries.
Use a table to figure out the SD
and the mean
n= 8
s = 5.33
 =244.5
d.f. =
7
table
tc =
3.499
E=
26.9
217.6 calories < μ < 271.4 calories
Resources
http://www.ntpu.edu.tw/stat/learning/people/gosset.htm
http://www.mrs.umn.edu/~sungurea/introstat/history/w98/gosset.html