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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

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 400-416, 425-428

Approximate Reading for Next Class:

Pages 431-439, 450-471

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.

Hypothesis Testing

Note: Can never make a definite conclusion,

Instead measure strength of evidence.

Reason: have to deal with uncertainty

But: Can quantify uncertainty

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

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

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-

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)

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?)

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?)

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?

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 a commonly agreed upon value, but very widely used:
- Drug testing
- Publication of scientific papers

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

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”

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)

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

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

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

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

>> ½, so no evidence at all for H-

(makes sense)

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)

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

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

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?

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 110. 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.

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- don’t dispute b. H- not safe

c. H- not safe d. H- safe)

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)

And now for somethingcompletely different….

An amazing movie clip:

http://abfhm.free.fr/basket.htm

Thanks to Trent Williamson

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

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

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

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

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

Hypothesis Testing

HW: 6.55, 6.61

Interpret with bothyes-no and gray level

AlternateTerminology:

“Significant at the 5% level” =

= P-value < 0.05

“Test Statistic z” = N(0,1) cutoff

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

Hypothesis Testing

Hypo Testing Approach III:

“Alternative Hypothesis” is:

HA = “H+ or H-”

General form: Specified Value

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”)

Hypothesis Testing, III

Basis of test:

(now see

why this distribution

form is

used)

observed value of

“more conclusive” is the two tailed area

Fast Food Business Example

Two Sided Viewpoint:

$1,000 $1,000

P-value =

$20,000 $21,000

mutually exclusive “or” rule

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

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

Hypothesis Testing, III

HW: 6.62 - interpret both yes-no & gray-level

(-2.20, 0.0278, rather strong evidence)

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

Hypothesis Testing, III

One-sided test of:

P-value = … = 0.031

Part 5 ofhttp://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls

Two-sided test of:

P-value = … = 0.062

Hypothesis Testing, III

Yes-no interpretation:

Have strong evidence

But no evidence !?!

(shouldn’t bigger imply different?)

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)

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

Hypothesis Testing, III

CAUTION: Read problem carefully to distinguish between:

One-sided Hypotheses - like:

Two-sided Hypotheses - like:

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….

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….

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”)

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

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?”

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

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).

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

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

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:

= prop’n of Females preferring English

= prop’n of Males preferring English

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