8-2 Estimation Estimating

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8-2 EstimationEstimating µ when s is UNKNOWN
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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!

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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 s. He needed another method
  to evaluate error.
• He worked with a very famous statistician, Karl
  Pearson, on understanding Standard Errors, and
  developed the Students T distribution

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Students 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.

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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 s. (not likely)
• Find t when you dont. (more likely)

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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

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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.

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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.

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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.
• Dont worry about the indication one tail and
  two tail. Well deal with that later.

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And?

• Just like yesterday,

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And?

• Just like yesterday,
• And
• Remember to use n 1 for d.f.

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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.

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Use a table to figure out the SD and the mean

• n
• s ? d.f.
• table
• tc
• E

8
244.5
7
5.33
3.499
26.9
217.6 calories lt µ lt 271.4 calories
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Resources

• http//www.ntpu.edu.tw/stat/learning/people/gosset
  .htm
• http//www.mrs.umn.edu/sungurea/introstat/history
  /w98/gosset.html