Stat 31, Section 1, Last Time

1
Stat 31, Section 1, Last Time

• Statistical Inference
• Confidence Intervals
• Range of Values to reflect uncertainty
• Bracket true value in 95 of repetitions
• Choice of sample size
• Choose n to get desired error
• Hypothesis Testing
• Yes No questions, under uncertainty

2
Reading In Textbook

• Approximate Reading for Todays Material
• Pages 400-416, 425-428
• Approximate Reading for Next Class
• Pages 431-439, 450-471

3
Hypothesis Tests

• E.g. A fast food chain currently brings in
  profits of 20,000 per store, per day. A new
  menu is proposed. Would it be more profitable?
• Test Have 10 stores (randomly selected!) try
  the new menu, let average of their daily
  profits.

4
Hypothesis Testing

• Note Can never make a definite conclusion,
• Instead measure strength of evidence.
• Reason have to deal with uncertainty
• But Can quantify uncertainty

5
Hypothesis Testing

• Approach I (note different from text)
• Choose among 3 Hypotheses
• H Strong evidence new menu is better
• H0 Evidence in inconclusive
• H- Strong evidence new menu is worse

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Caution!!!

• Not following text right now
• This part of course can be slippery
• I am breaking this down to basics
• Easier to understand
• (If you pay careful attention)
• Will tie things together later
• And return to textbook approach later

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Fast Food Business Example

• Base decision on best guess
• Will quantify strength of the evidence using
  probability distribution of
• E.g. ? Choose H
• ? Choose
  H0
• ? Choose
  H-

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Fast Food Business Example

• How to draw line?
• (There are many ways,
• here is traditional approach)
• Insist that H (or H-) show strong evidence
• I.e. They get burden of proof
• (Note one way of solving
• gray area problem)

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Fast Food Business Example

• Suppose observe ,
• based on
• Note , but is this
  conclusive?
• or could this be due to natural sampling
  variation?
• (i.e. do we risk losing money from new menu?)

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Fast Food Business Example

• Assess evidence for H by
• H p-value Area

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Fast Food Business Example

• Computation in EXCEL
• Class Example 22, Part 1
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg24.xls
• P-value 0.094
• i.e. About 10
• Is this small?
• (where do we draw the line?)

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Fast Food Business Example

• View 1 Even under H0, just by chance, see
  values like , about 10 of
  the time,
• i.e. 1 in 10,
• so not terribly convincing???
• Could be a fluke?
• But where is the boundary line?

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P-value cutoffs

• View 2 Traditional (and even legal) cutoff,
  called here the yes-no cutoff
• Say evidence is strong,
• when P-value lt 0.05
• Just a commonly agreed upon value, but very
  widely used
• Drug testing
• Publication of scientific papers

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P-value cutoffs

• Say results are statistically significant when
  this happens, i.e. P-value lt 0.05
• Can change cutoff value 0.05, to some other
  level, often called
• Greek alpha
• E.g. your airplane safe to fly,
• want
• E.g. often called strongly
  significant

15
P-value cutoffs

• View 3 Personal idea about cutoff,
• called gray level (vs. yes-no above)
• P-value lt 0.01 quite strong evidence
• 0.01 lt P-value lt 0.1 weaker evidence
• but stronger for smaller
  P-val.
• 0.1 lt P-value very weak evidence, at
• 
  best

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Gray Level Cutoffs

• View 3 gray level (vs. yes-no above)
• Note only about interpretation of P-value
• E.g. When P-value is given
• HW 6.40 (d) give gray level interp.
• (no, no, relatively weak evidence)
• 6.41 (d) give gray level interp.
• (yes, not, moderately strong evidence)

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Caution!!!

• Gray level viewpoint not in text
• Will see it is more sensible
• Hence I teach this
• Suggest you use this later in life
• Will be on HW exams

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Fast Food Business Example

• P-value of 0.094 for H,
• Is quite weak evidence for H,
• i.e. only a mild suggestion
• This happens sometimes not enough information
  in data for firm conclusion

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Fast Food Business Example

• Flip side could also look at strength of
  evidence for H-.
• Expect very weak, since saw
• Quantification
• H- P-value 20,000
  21,000

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Fast Food Business Example

• EXCEL Computation
• Class Example 24, part 1
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg24.xls
• H- P-value 0.906
• gtgt ½, so no evidence at all for H-
• (makes sense)

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Fast Food Business Example

• A practical issue
• Since ,
• May want to gather more data
• Could prove new menu clearly better
• (since more data means more
• information, which
• could overcome uncertainty)

22
Fast Food Business Example

• Suppose this was done, i.e. n 10 is replaced
  by n 40, and got the same
• Expect 4 times the data ? ½ of the SD
• Impact on P-value?
• Class Example 24, Part 2
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg24.xls

23
Fast Food Business Example

• How did it get so small, with only ½ the SD?
• mean 20,000,
  observed 21,000
• P-value 0.094
  P-value 0.004

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Hypothesis Testing

• HW C20
• For each of the problems
• A box label claims that on average boxes contain
  40 oz. A random sample of 12 boxes shows on
  average 39 oz., with s 2.2. Should we dispute
  the claim?

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Hypothesis Testing

1. We know from long experience that Farmer As pigs
   average 570 lbs. A sample of 16 pigs from Farmer
   B averages 590 lbs, with an SD of 110. Is it
   safe to say Bs pigs are heavier on average?
2. Same as (b) except lighter on average.
3. Same as (b) except that Bs average is 630 lbs.

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Hypothesis Testing

• Do
• Define the population mean of interest.
• Formulate H, H0, and H-, in terms of mu.
• Give the P-values for both H and H-.
• (a. 0.942, 0.058, b. 0.234, 0.766,
• c. 0.234, 0.766, d. 0.015, 0.985)
• Give a yes-no answer to the questions.
• (a. H- ? dont dispute b. H- ? not safe
• c. H- ? not safe d. H- ? safe)

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Hypothesis Testing

• Give a gray level answer to the questions.
• (a. H- ? moderate evidence against
• b. H- ? no strong evidence
• c. H- ? seems to go other way
• d. H- ? strong evidence, almost very strong)

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And now for somethingcompletely different.

• An amazing movie clip
• http//abfhm.free.fr/basket.htm
• Thanks to Trent Williamson

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Hypothesis Testing

• Hypo Testing Approach II
• 1-sided testing
• (more conventional is version in text)
• Idea only one of H and H- is usually relevant,
  so combine other with H0

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Attention!!!

• Now return to textbook presentation
• H-, H0, and H ideas are building blocks
• Will combine these
• In two different ways
• To get more conventional hypothesis
• As developed in text

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Hypothesis Testing

• Approach II New Hypotheses
• Null Hypothesis H0 H0 or
• Alternate Hypothesis HA opposite of
• Note common notation for HA is H1
• Gets burden of proof, I might
  accidentally put this
• i.e. needs strong evidence to prove this

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Hypothesis Testing

• Weird terminology Firm conclusion is called
  rejecting the null hypothesis
• Basics of Test P-value
• Note same as H0 in H, H0, H- case,
• so really just same as above

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Fast Food Business Example

• Recall New menu more profitable???
• Hypo testing setup
• P-val
• Same as before.
• See Class Example 24, part 3
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg24.xls

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Hypothesis Testing

• HW 6.55, 6.61
• Interpret with both yes-no and gray level
• AlternateTerminology
• Significant at the 5 level
• P-value lt
  0.05
• Test Statistic z N(0,1) cutoff

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Hypothesis Testing

• Hypo Testing Approach III
• 2-sided tests
• Main idea when either of H or H- is
  conclusive, then combine them
• E.g. Is population mean equal to a given value,
  or different?
• Note either bigger or smaller is strong evidence

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Hypothesis Testing

• Hypo Testing Approach III
• Alternative Hypothesis is
• HA H or H-
• General form Specified Value

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Hypothesis Testing, III

• Note always goes in HA, since cannot
  have strong evidence of .
• i. e. cannot be sure about difference between
  and 0.000001
• while can have convincing evidence for
• (recall HA gets burden of proof)

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Hypothesis Testing, III

• Basis of test
• (now see
• why this
  distribution
• form is
• used)
• 
  observed value of
• more conclusive is the two tailed area

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Fast Food Business Example

• Two Sided Viewpoint
• 1,000
  1,000
• P-value
• 
  20,000 21,000
• mutually exclusive
  or rule

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Fast Food Business Example

• P-value
• NORMDIST
• See Class Example 24, part 4
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg24.xls
• 0.188
• So no strong evidence,
• Either yes-no or gray-level

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Fast Food Business Example

• Shortcut by symmetry
• 2 tailed Area
  2 x Area
• See Class Example 24, part 4
• http//stat-or.unc.edu/webspace/postscript/marron/
  Teaching/stor155-2007/Stor155Eg24.xls

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Hypothesis Testing, III

• HW 6.62 - interpret both yes-no
  gray-level
• (-2.20, 0.0278, rather strong evidence)

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Hypothesis Testing, III

• A paradox of 2-sided testing
• Can get strange conclusions
• (why is gray level sensible?)
• Fast food example suppose gathered more data,
  so n 20, and other results are the same

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Hypothesis Testing, III

• One-sided test of
• P-value 0.031
• Part 5 of http//stat-or.unc.edu/webspace/posts
  cript/marron/Teaching/stor155-2007/Stor155Eg24.xls
  
• Two-sided test of
• P-value 0.062

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Hypothesis Testing, III

• Yes-no interpretation
• Have strong evidence
• But no evidence !?!
• (shouldnt bigger imply different?)

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Hypothesis Testing, III

• Notes
• Shows that yes-no testing is different from usual
  logic
• (so be careful with it!)
• Reason 2-sided admits more uncertainty into
  process
• (so near boundary could make
• a difference, as happened here)
• Gray level view avoids this
• (1-sided has stronger evidence,
• as expected)

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Hypothesis Testing, III

• Lesson 1-sided vs. 2-sided issues need
  careful
• Implementation
• (choice does affect answer)
• Interpretation
• (idea of being tested
• depends on this choice)
• Better from gray level viewpoint

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Hypothesis Testing, III

• CAUTION Read problem carefully to distinguish
  between
• One-sided Hypotheses - like
• Two-sided Hypotheses - like

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Hypothesis Testing

• Hints
• Use 1-sided when see words like
• Smaller
• Greater
• In excess of
• Use 2-sided when see words like
• Equal
• Different
• Always write down H0 and HA
• Since then easy to label more conclusive
• And get partial credit.

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Hypothesis Testing

• E.g. Text book problem 6.34
• In each of the following situations, a
  significance test for a population mean, is
  called for. State the null hypothesis, H0 and
  the alternative hypothesis, HA in each case.

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Hypothesis Testing

• E.g. 6.34a
• An experiment is designed to measure the effect
  of a high soy diet on bone density of rats.
• Let
• average bone density of high soy rats
• average bone density of ordinary rats
• (since no question of bigger or smaller)

52
Hypothesis Testing

• E.g. 6.34b
• Student newspaper changed its format. In a
  random sample of readers, ask opinions on scale
  of -2 new format much worse, -1 new format
  somewhat worse, 0 about same, 1 new a
  somewhat better, 2 new much better.
• Let
• average opinion score

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Hypothesis Testing

• E.g. 6.34b (cont.)
• No reason to choose one over other, so do two
  sided.
• Note Use one sided if question is of form is
  the new format better?

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Hypothesis Testing

• E.g. 6.34c
• The examinations in a large history class are
  scaled after grading so that the mean score is
  75. A teaching assistant thinks that his
  students have a higher average score than the
  class as a whole. His students can be considered
  as a sample from the population of all students
  he might teach, so he compares their score with
  75.
• average score for all students of this TA

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Hypothesis Testing

• E.g. Textbook problem 6.36
• Translate each of the following research
  questions into appropriate and
• Be sure to identify the parameters in each
  hypothesis (generally useful, so already did this
  above).

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Hypothesis Testing

• E.g. 6.36a
• A researcher randomly divides 6-th graders into 2
  groups for PE Class, and teached volleyball
  skills to both. She encourages Group A, but acts
  cool towards Group B. She hopes that
  encouragement will result in a higher mean test
  for group A.
• Let
• mean test score for Group A
• mean test score for Group B

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Hypothesis Testing

• E.g. 6.36a
• Recall Set up point to be proven as HA

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Hypothesis Testing

• E.g. 6.36b
• Researcher believes there is a positive
  correlation between GPA and esteem for students.
  To test this, she gathers GPA and esteem score
  data at a university.
• Let
• correlation between GPS esteem

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Hypothesis Testing

• E.g. 6.36c
• A sociologist asks a sample of students which
  subject they like best. She suspects a higher
  percentage of females, than males, will name
  English.
• Let
• propn of Females preferring English
• propn of Males preferring English

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Hypothesis Testing

• HW on setting up hypotheses
• 6.35, 6.37