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Stor 155, Section 2, Last Time • Hypothesis Testing – Assess strength of evidence with P- value • P-value interpretation: – Yes – No – Gray – level – 1 - sided vs. 2 - sided “paradox” bdry H H conclusive more or saw what P value P A & | 0

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Page 1: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Stor 155, Section 2, Last Time• Hypothesis Testing

– Assess strength of evidence with P-value

• P-value interpretation:

– Yes – No

– Gray – level

– 1 - sided vs. 2 - sided “paradox”

bdryHHconclusivemoreorsawwhatPvalueP A&| 0

Page 2: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 400-416, 424-428, 450-471

Approximate Reading for Next Class:

Pages 485-504

Page 3: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

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

Page 4: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing, III

One-sided test of:

P-value = … = 0.031

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

Two-sided test of:

P-value = … = 0.062

000,20$:0 H

000,20$: AH

000,20$:0 H

000,20$: AH

https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg22.xls
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls
Page 5: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing, III

Yes-no interpretation:

Have strong evidence

But no evidence !?!

(shouldn’t bigger imply different?)

000,20$

000,20$

Page 6: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing, IIINotes:i. Shows that yes-no testing is different

from usual logic(so be careful with it!)

ii. Reason: 2-sided admits more uncertainty into process

(so near boundary could make a difference, as happened here)

iii. Gray level view avoids this:(1-sided has stronger evidence,

as expected)

Page 7: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing, III

Lesson: 1-sided vs. 2-sided issues need careful:

1. Implementation

(choice does affect answer)

2. Interpretation

(idea of being tested

depends on this choice)

Better from gray level viewpoint

Page 8: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing, III

CAUTION: Read problem carefully to distinguish between:

One-sided Hypotheses - like:

Two-sided Hypotheses - like:

:.:0 AHvsH

:.:0 AHvsH

Page 9: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingHints:• 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….

Page 10: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

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….

Page 11: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingE.g. 6.34aAn 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”)

O

OHSH :0OHSAH :

HS

Page 12: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingE.g. 6.34bStudent 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

Page 13: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingE.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?”

0:0 H

0: AH

Page 14: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingE.g. 6.34cThe 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 TA75:0 H 75: AH

Page 15: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

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).

0H AH

Page 16: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingE.g. 6.36aA 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 BAB

Page 17: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingE.g. 6.36a

Recall: Set up point to be proven as HA

BAH :0

BAAH :

Page 18: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingE.g. 6.36bResearcher 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

0:0 H

0: AH

Page 19: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis TestingE.g. 6.36cA 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

Fp

MF ppH :0

MFA ppH :

Mp

Page 20: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

HW on setting up hypotheses:

6.35, 6.37

Page 21: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

Connection between Confidence Intervals

and Hypothesis Tests:

Reject at Level 0.05 P-value < 0.05

dist’n

Area < 0.05

0.95 margin of error

mX

mXmX , CIinnot 95.0

X

Page 22: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing & CIs

Reject at Level 0.05

Notes:

1. This is why EXCEL’s CONFIDENCE function uses = 1 – coverage prob.

2. If only care about 2-sided hypos, then could work only with CIs

(and not learn about hypo. tests)

CIinnot 95.0

Page 23: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing & CIs

HW: 6.71

Page 24: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

The three traps of Hypothesis Testing

(and how to avoid them…)

Trap 1: Statistically Significant is different

from Really Significant

(don’t confuse them)

Page 25: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing Traps

Trap 1: Statistically Significant is different

from Really Significant

E.g. To test a painful diet program, 10,000

people were put on it. Their average

weight loss was 1.7 lbs, with s = 73.

Assess “significance” by hypothesis testing.

Page 26: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing Traps

Trap 1: Statistically Significant is different

from Really Significant

See Class Example 25: Trap 1http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg25.xls

P-value = 0.0099

Strongly Statistically

Significant

Careful: Is this practically significant?

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg25.xls
Page 27: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing Traps

Trap 1, e.g: Is this practically significant?

NO! Not worth painful diet to lose 1.7 lbs.

Resolution: Hypo. testing resolves question:

Could observed results be due to

chance variation?

Answer here is no, since n is really large.

Page 28: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing Traps

Trap 1, e.g: Is this practically significant?

Answer here is no, since n is really large.

But this is different from question:

Do results show a big difference?

Page 29: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing Traps

Trap 2: Insignificant results do not mean

nothing is there,

Only: Didn’t have strong enough data to

actually prove results.

E.g. Class 25, Trap 2http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg25.xls

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg25.xls
Page 30: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing Traps

Trap 3: Try enough tests, and you will find “something” even where it doesn’t exist.

Revisit Class Example 21, Q4http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg23.xls

We saw about 5% of CIs don’t cover.

So, (using CI – Hypo Test connection), expect about 5% of tests to choose HA, (and claim

“strong evidence) even when H0 true.

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg23.xls
Page 31: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing Traps

Strategies to avoid Trap 3:

1. Scientific Method:

Form Hypothesis tests once.

2. For repeated tests: use careful adjustments:

(beyond scope of this course)

Get help if needed

Page 32: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing Traps

HW: 6.74 (about 50)

6.82

(0.382, 0.171, 0.0013)

6.83, 6.84

(0.0505, 0.0495)

Page 33: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Some stuff for next time

(next slides)

Page 34: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

View 3: level testing

Idea: instead of reporting P-value, choose a

fixed level, say 5%

Then reject H0, i.e. find strong evidence…

When P-value < 0.05 (more generally )

(slight recasting of yes-no version of testing)

HW: 6.53 (careful, already assigned above)

Page 35: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

View 4: P-value shows only half of the

decision problem:

Graphical Illustration: Truth

H0 true HA true

Test Result: Choose H0

Choose HA

Correct Type II Error

Type I Error

Correct

Page 36: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

View 4: Both sides of decision problem:

Small P-values Small Type I error

What about Type II error?

(seems part of problem)

a. Simplistic Answer: Don’t care because

have put burden of proof on HA.

Page 37: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

View 4: What about Type II error?

b. Deeper Answer: Does matter for test

sensitivity issues, and test power issues

E.g. how large a sample size is needed for a

given test power (treated above)?

Page 38: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

View 4: Terminology:

P{Type I Error} is called level of test

1 - P{Type I Error} is called specificity

1 – P{Type II Error} is called power of test

1 – P{Type II Error} is called sensitivity

Page 39: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

A statistician’s view on politics…

Some Current Controversial Issues:

• North Carolina State Lottery

• Replace Social Security by Individual Retirement Plans

Debate is passionate, (natural for complex and important issues)

But what is missing?

Page 40: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Review Ideas on State Lotteries,

from our study of Expected Value

Not an obvious choice because:

• Gambling is (at least) unsavory:– Religious objections– Some like it too much– Destroys some lives

Page 41: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

State Lotteries, not an obvious choice:• The only totally voluntary tax:

– Nobody required, unlike all other taxes– Money often used for education– Good or bad, given state of economy???

• Highest tax burden on the poor– Poor enjoy playing much more– Higher taxes on poor better for society???– Tendency towards “rich get richer”???

Page 42: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

What about Individual Retirement Plans:

Main Benefit:

On average individual investments

return greater yields

than government investments

So can we conclude:

• “Overall we are all better off”???

• Since more total money to go around?

Page 43: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Very common mistake in this reasoning:

• Notice “on average” part of statement

• Should also think about variation about

the average???

Page 44: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Variation about average Issue 1:

• Should think of population of people

• Average is over this population

• Except some to do great

• And expect some to lose everything

• What will the percentage of losers be?

• What do we do with those who lose all?

• What will that cost?

Page 45: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Variation about average Issue 2:

• Also are averaging over time

• Overall gains of stock market happen only over this average

• Some need $$$ when market is down

• How often will this happen?

• How do we deal with it?

Page 46: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Main concept I hope you carry away from this course:

Variation is a fundamental concept

• Look for it

• Think about it

• Ask questions about it

(Vital to informed citizenship)

Page 47: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Australian joke about Variation:

Did you hear about the man who drowned in a lake with average depth 6 inches?

Page 48: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Australian joke about Variation:

He understood “average”, but not variation about the average

Page 49: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Really have such lakes?

Yes, in Australia

Page 50: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

And now for something completely different…

Suggestions of such issues (politics, controversy…) for discussion are welcome….

Page 51: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing

Other views of hypothesis testing:

View 2: Z-scores

Idea: instead of reporting p-value (to assess statistical significance)

Report the Z-score

A different way of measuring significance

Page 52: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing – Z scores

E.g. Fast Food Menus:

Test

Using

P-value = P{what saw or m.c.| H0 & HA bd’ry}

000,20$:0 H

000,20$: AH

10,400,2$,000,21$ nsX

Page 53: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing – Z scores

P-value = P{what saw or or m.c.| H0 & HA bd’ry}

rybdXP '|000,21$

000,20$|000,21$ XP

102400$

000,20$000,21$

nsX

P

317.1 ZP

Page 54: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing – Z scores

P-value

This is the Z-score

Computation: Class E.g. 24, Part 6http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls

Distribution: N(0,1)

317.1 ZP

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls
Page 55: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Hypothesis Testing – Z scores

P-value

So instead of reporting tail probability,

Report this cutoff instead,

as “SDs away from mean $20,000”

HW: 6.67 , but use NORMDIST, not Table D

317.1 ZP

Page 56: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Sec. 7.1: Deeper look at Inference

Recall: “inference” = CIs and Hypo Tests

Main Issue: In sampling distribution

Usually is unknown, so replace with an estimate, .

For n large, should be “OK”, but what about:

• n small?

• How large is n “large”?

nNX /,0~

s

Page 57: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Unknown SD

Approach: Account for “extra variability in the approximation”

Mathematics: Assume individual

I.e.

• Data have mound shaped histogram

• Recall averages generally normal

• But now must focus on individuals

s ,~ NX i

Page 58: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

Unknown SD

Then

Replace by , then

has a distribution named:

“t-distribution with n-1 degrees of freedom”

nNX /,~

1,0~ N

n

X

sn

sX

Page 59: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

t - Distribution

Notes:

1. n is a parameter (like ) that controls “added variability from approximation”

View: Study Densities,

over degrees of freedom…http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/EgTDist.mpg

,,, ps

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/EgTDist.mpg
Page 60: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

t - Distribution

Comments on movie

• Serious differences for n <= 10

(since approximation is bad)

• Very little difference for n large

(since approximation really good)

• Rule of thumb cutoff: many say

“negligible difference for n > 30”

• Cutoffs more informative than curves

s

s

Page 61: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

t - Distribution

Nice Alternate View:Webster West (U. So. Carolina) Applet:

http://www.stat.sc.edu/~west/applets/tdemo1.html

Notes:– Big change for small degrees of freedom– Looks very normal for large degrees of

freedom– I would overlay with Normal…

http://www.stat.sc.edu/~west/applets/tdemo1.html
Page 62: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

t - Distribution

Notes:

2. Careful: set “degrees of freedom” =

= n – 1 (not n)

• Easy to forget later

• Good to add to sheet of notes for exam

HW: 7.21 a

Page 63: Stor 155, Section 2, Last Time Hypothesis Testing –Assess strength of evidence with P-value P-value interpretation: –Yes – No –Gray – level –1 - sided

t - Distribution

Notes:

3. Must work with standardized version of

i.e.

• No longer can plug mean and SD

• into EXCEL formulas

• In text this was already done,

• Since need this for Normal table calc’ns

nsX X


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