stat 31, section 1, last time hypothesis tests –h -, h 0 or h + version of testing –p-values –...

Stat 31, Section 1, Last Time

• Hypothesis Tests

– H-, H0 or H+ version of testing

– P-values – small for strong evidence

– 1-sided (strong evidence for “>” or “<“)

– 2-sided (strong evidence for “different”)

• Cutoffs

– Yes – no (P-val < 0.05, or not?)

– Gray level (interpret level…)

Hypothesis Testing, III

CAUTION: Read problem carefully to distinguish between:

One-sided Hypotheses - like:

Two-sided Hypotheses - like:

:.:0 AHvsH

:.:0 AHvsH

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….

Hypothesis Testing

E.g. Text book problem 6.32:

In each of the following situations, a

significance test for a population mean,

mu is called for. State the null

hypothesis, H0 and the alternative

hypothesis, HA in each case….

Hypothesis TestingE.g. 6.32aExperiments on learning in animals

sometimes measure how long it takes a mouse to find its way through mazes. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the nice to complete the maze faster. She measures how long each of 10 mice takes with a noise as stimulus.

= time to finish with loud noise.18:0 H 18: AH

Hypothesis TestingE.g. 6.32bThe examinations in a large history class are

scaled after grading so that the mean score is 50. 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 50.

= average score for all students of this TA50:0 H 50: AH

Hypothesis TestingE.g. 6.32cThe Census Bureau reports that households

spend an average of 31% of their total spending on housing. A homebuilders association in Cleveland wonders if the national finding applies in their area. They interview a sample of 40 households in the Cleveland metropolitan area to learn what percent of their spending goes toward housing.

= avg. % for housing in Cleveland %31:0 H %31: AH

Hypothesis Testing

E.g. Textbook problem 6.34

Translate each of the following research

questions into appropriate and0H AH

Hypothesis TestingE.g. 6.34aCensus Bureau data shows that the mean

household income in the area served by a shopping mall is $62,500 per year. A market research firm questions shoppers at the mall to find out whether the mean household income of mall shoppers is higher that that of the general population.

= average income of mall shoppers 500,62$:0 H

500,62$: AH

Hypothesis TestingE.g. 6.34bLast year, your company’s service

technicians took an average of 2.6 hours to respond to trouble calls from business customers who had purchased service contracts. Do this year’s data show a different average response size?

= average response time 6.2:0 H6.2: AH

Hypothesis Testing

HW on setting up hypotheses:

6.31, 6.33

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

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

Hypothesis Testing & CIs

HW: 6.63, 6.64

(a. [59.99,63.59] b. No, 61.3 inside c. No, 63 inside)

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)

Hypothesis Testing Traps



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.




See Class Example 23: Trap 1https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg23.xls

P-value = 0.0099

Strongly Statistically Significant

Careful: Is this practically significant?

https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg23.xls


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.


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?


Trap 2: Insignificant results do not mean

nothing is there,

Only: Didn’t have strong enough data to

actually prove results.

E.g. Class 23, Trap 2https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg23.xls



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

Revisit Class Example 21, Q4https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg21.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.



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


HW: 6.69, 6.71, 6.73, 6.75, 6.77

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?


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


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”???


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?


Very common mistake in this reasoning:

• Notice “on average” part of statement

• Should also think about variation about

the average???


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?


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?


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)


Australian joke about Variation:

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


Australian joke about Variation:

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


Really have such lakes?

Yes, in Australia


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

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

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


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


P-value

This is the Z-score

Computation: Class E.g. 22, Part 6https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg22.xls

Distribution: N(0,1)

317.1 ZP



P-value

So instead of reporting tail probability,

Report this cutoff instead,

as “SDs away from mean $20,000”

HW: 6.59 , but use NORMDIST, not Table D

317.1 ZP

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)

stat 31, section 1, last time hypothesis tests –h -, h 0 or h + version of testing –p-values –...

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