8-2 estimation estimating μ when σ is unknown. imagine you are in charge of quality control at...

8-2 EstimationEstimating 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

With a new parameter, called degrees of freedom (d.f.) which = n – 1.

x μt

sn

Compares to z?

Symmetric about μ = 0. Bell Shaped.

Difference? The tails are a bit higherYou 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 valueThe 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,

And

Remember to use n – 1 for d.f.

cσE t

n

x E μ x E

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 287Use 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 = s = = d.f. = table

tc =

E =

8

7

3.499

5.33 244.5

26.9

217.6 calories < μ < 271.4 calories

Resourceshttp://www.ntpu.edu.tw/stat/learning/people/gosset.htm

http://www.mrs.umn.edu/~sungurea/introstat/history/w98/gosset.html