1. Stat 31, Section 1, Last Time Choice of sample size
Choose n to get desired error
Interpretation of Confidence Intervals
Bracket true value in 95% of repetitions
Hypothesis Testing
Yes – No questions, under uncertainty

2. 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.

3. Hypothesis Testing Note: Can never make a definite conclusion,
Instead measure strength of evidence.
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

4. 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-

5. 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)

6. 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?)

7. Fast Food Business Example
Assess evidence for H+ by:
H+ p-value = Area

8. Fast Food Business Example
Computation in EXCEL:
Class Example 22, Part 1:
P-value = 0.094
i.e. About 10%
Is this “small”?
(where do we draw the line?)

9. 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?

10. P-value cutoffs
View 2: Traditional (and even “legal”) cutoff, called here the yes-no cutoff:
Say evidence is strong,
when P-value < 0.05
Just an agreed upon value, but very widely used:
Drug testing
Publication of scientific papers

11. P-value cutoffs
Say “results are statistically significant” when this happens, i.e. P-value < 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

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

13. Fast Food Business Example
P-value of for H+,
Is “quite weak evidence for H+”,
i.e. “only a mild suggestion”
This happens sometime: not enough information in data for firm conclusion

14. 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

15. Fast Food Business Example
EXCEL Computation:
Class Example 22, part 1
H- P-value = 0.906
>> ½, so no evidence at all for H-
(makes sense)

16. 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)

17. 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 22, Part 2

18. Fast Food Business Example
How did it get so small, with only ½ the SD?
mean = $20,000, observed $21,000
P-value = P-value = 0.004

19. Hypothesis Testing HW: C16 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 = Should we dispute the claim?

20. Hypothesis Testing
We know from long experience that Farmer A’s pigs average 570 lbs. A sample of 16 pigs from Farmer B averages 590 lbs, with an SD of Is it safe to say B’s pigs are heavier on average?
Same as (b) except “lighter on average”.
Same as (b) except that B’s average is 630 lbs.

21. 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 , 0.058, b , 0.766,
c , 0.766, d , 0.985)
Give a yes-no answer to the questions.
(a. H-  don’t dispute b. H-  not safe
c. H-  not safe d. H-  safe)

22. 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)

23. And now for something completely different….
An amazing movie clip:
Thanks to Trent Williamson

24. 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

25. 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

26. 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

27. Fast Food Business Example
Recall: New menu more profitable???
Hypo testing setup:
P-val =
Same as before.
See: Class Example 22, part 3:

28. Hypothesis Testing HW: 6.45, 6.53
Interpret with both yes-no and gray level
AlternateTerminology:
“Significant at the 5% level” =
= P-value < 0.05
“Test Statistic z” = N(0,1) cutoff

29. Hypothesis Testing 2-sided tests
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

30. Hypothesis Testing “Alternative Hypothesis” is: HA = “H+ or H-”
Hypo Testing Approach III:
“Alternative Hypothesis” is:
HA = “H+ or H-”
General form: Specified Value

31. Hypothesis Testing “Alternative Hypothesis” is: HA = “H+ or H-”
Hypo Testing Approach III:
“Alternative Hypothesis” is:
HA = “H+ or H-”
General form: Specified Value

32. Hypothesis Testing, III
Note: “ ” always goes in HA, since cannot have “strong evidence of =”.
i. e. cannot be sure about difference between and
while can have convincing evidence for “ ”
(recall H0 gets “burden of proof”)

33. Hypothesis Testing, III
Basis of test:
(now see
why this distribution
form is
used)
observed value of
“more conclusive” is the two tailed area

34. Fast Food Business Example
Two Sided Viewpoint:
$1, $1,000
P-value =
$20, $21,000
mutually exclusive “or” rule

35. Fast Food Business Example
P-value =
=NORMDIST…
See Class Example 22, part 4
= 0.188
So no strong evidence,
Either yes-no or gray-level

36. Fast Food Business Example
Shortcut: by symmetry
2 tailed Area = 2 x Area
See Class Example 22, part 4

37. Hypothesis Testing, III
HW: interpret both yes-no & gray-level
(0.0164, quite strong evidence, OK for non-normal, by Central Limit Theorem)

38. Hypothesis Testing, III
A “paradox” of 2-sided testing:
Can get strange conclusions
Fast food example: suppose gathered more data, so n = 20, and other results are the same

39. Hypothesis Testing, III
One-sided test of:
P-value = … = 0.031
Part 5 of
Two-sided test of:
P-value = … = 0.062

40. Hypothesis Testing, III
Yes-no interpretation:
Have strong evidence
But no evidence !?!
(shouldn’t bigger imply different?)

41. 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)

42. 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

43. Hypothesis Testing, III
CAUTION: Read problem carefully to distinguish between:
One-sided Hypotheses - like:
Two-sided Hypotheses - like: