STATISTICAL INFERENCES

CHAPTERS 12 AND 13

•HYPOTHESIS TESTING FOR PROPORTIONS AND MEANS

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2

TESTING HYPOTHESES ABOUT PROPORTIONS

• PROBLEM• SUPPOSE WE TOSSED A COIN 100 TIMES

AND WE OBTAINED 38 HEADS AND 62 TAILS. IS THE COIN BIASED?

• THERE IS NO WAY TO SAY YES OR NO WITH 100% CERTAINTY. BUT WE MAY EVALUATE THE STRENGTH OF SUPPORT TO THE HYPOTHESIS THAT “THE COIN IS BIASED.”

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TESTING

• HYPOTHESESNULL HYPOTHESIS – ESTABLISHED FACT;– A STATEMENT THAT WE EXPECT DATA TO

CONTRADICT;– NO CHANGE OF PARAMETERS.ALTERNATIVE HYPOTHESIS – NEW CONJECTURE;– YOUR CLAIM;– A STATEMENT THAT NEEDS A STRONG

SUPPORT FROM DATA TO CLAIM IT;– CHANGE OF PARAMETERS

0H

AH

4

IN OUR PROBLEM

.""

5.0;:

5.0;:0

HEADSTURNSCOINTHE

THATYPROBABILITTHEISpWHERE

pBIASEDISCOINH

pFAIRISCOINH

A

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EXAMPLE

• WRITE THE NULL AND ALTERNATIVE HYPOTHESES YOU WOULD USE TO TEST EACH OF THE FOLLOWING SITUATIONS.

• (A) IN THE 1950s ONLY ABOUT 40% OF HIGH SCHOOL GRADUATES WENT ON TO COLLEGE. HAS THE PERCENTAGE CHANGED?

• (B) 20% OF CARS OF A CERTAIN MODEL HAVE NEEDED COSTLY TRANSMISSION WORK AFTER BEING DRIVEN BETWEEN 50,000 AND 100,000 MILES. THE MANUFACTURER HOPES THAT REDESIGN OF A TRANSMISSION COMPONENT HAS SOLVED THIS PROBLEM.

• (C) WE FIELD TEST A NEW FLAVOR SOFT DRINK, PLANNING TO MARKET IT ONLY IF WE ARE SURE THAT OVER 60% OF THE PEOPLE LIKE THE FLAVOR.

6

ATTITUDE

• ASSUME THAT THE NULL HYPOTHESIS

IS TRUE AND UPHOLD IT,

UNLESS DATA STRONGLY SPEAKS AGAINST IT.

0H

7

TEST MECHANIC

• FROM DATA, COMPUTE THE VALUE OF A PROPER TEST STATISTICS, THAT IS, THE Z-STATISTICS.

• IF IT IS FAR FROM WHAT IS EXPECTED UNDER THE NULL HYPOTHESIS ASSUMPTION, THEN WE REJECT THE NULL HYPOTHESIS.

8

COMPUTATION OF THE Z – STATISTICS OR PROPER TEST STATISTICS

n

qppSD

wherepSD

ppz

oo

o

.)ˆ(

,)ˆ(

ˆ

9

CONSIDERING THE EXAMPLE AT THE BEGINNING:

4.205.0

50.038.0

05.0100

)5.0(5.0)ˆ(,5.0,38.0ˆ

o

O

zAND

PSDPP

10

THE P – VALUE AND ITS COMPUTATION

• THE PROBABILITY THAT IF THE NULL HYPOTHESIS IS CORRECT, THE TEST STATISTIC TAKES THE OBSERVED OR MORE EXTREME VALUE.

• P – VALUE MEASURES THE STRENGTH OF EVIDENCE AGAINST THE NULL HYPOTHESIS. THE SMALLER THE P – VALUE, THE STRONGER THE EVIDENCE AGAINST THE NULL HYPOTHESIS.

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THE WAY THE ALTERNATIVE HYPOTHESIS IS WRITTEN IS HELPFUL IN COMPUTING THE P - VALUE

NORMAL CURVEAH

oA ppH :

oA ppH :

)( ozzP

)( ozzP

)(2 ozzP oA ppH :

valuep

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IN OUR EXAMPLE,

• P – VALUE = P( z < - 2.4) = 0.0082

• INTERPRETATION: IF THE COIN IS FAIR, THEN THE PROBABILITY OF OBSERVING 38 OR FEWER HEADS IN 100 TOSSES IS 0.0082

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CONCLUSION: GIVEN SIGNIFICANCE LEVEL = 0.05

• WE REJECT THE NULL HYPOTHESIS IF THE P – VALUE IS LESS THAN THE SIGNIFICANCE LEVEL OR ALPHA LEVEL.

• WE FAIL TO REJECT THE NULL HYPOTHESIS (I.E. WE RETAIN THE NULL HYPOTHESIS) IF THE P – VALUE IS GREATER THAN THE SIGNIFICANCE LEVEL OR ALPHA LEVEL.

14

ASSUMPTIONS AND CONDITIONS

• RANDOMIZATION

• INDEPENDENT OBSERVATIONS

• 10% CONDITION

• SUCCESS/FAILURE CONDITION

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EXAMPLE 1

• THE NATIONAL CENTER FOR EDUCATION STATISTICS MONITORS MANY ASPECTS OF ELEMENTARY AND SECONDARY EDUCATION NATIONWIDE. THEIR 1996 NUMBERS ARE OFTEN USED AS A BASELINE TO ASSESS CHANGES. IN 1996, 31% OF STUDENTS REPORTED THAT THEIR MOTHERS HAD GRADUATED FROM COLLEGE. IN 2000, RESPONSES FROM 8368 STUDENTS FOUND THAT THIS FIGURE HAD GROWN TO 32%. IS THIS EVIDENCE OF A CHANGE IN EDUCATION LEVEL AMONG MOTHERS?

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EXAMPLE 1 CONT’D

• (A) WRITE APPROPRIATE HYPOTHESES.

• (B) CHECK THE ASSUMPTIONS AND CONDITIONS.

• (C) PERFORM THE TEST AND FIND THE P – VALUE.

• (D) STATE YOUR CONCLUSION.

• (E) DO YOU THINK THIS DIFFERENCE IS MEANINGFUL? EXPLAIN.

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SOLUTION

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EXAMPLE 2

• IN THE 1980s IT WAS GENERALLY BELIEVED THAT CONGENITAL ABNORMALITIES AFFECTED ABOUT 5% OF THE NATION’S CHILDREN. SOME PEOPLE BELIEVE THAT THE INCREASE IN THE NUMBER OF CHEMICALS IN THE ENVIRONMENT HAS LED TO AN INCREASE IN THE INCIDENCE OF ABNORMALITIES. A RECENT STUDY EXAMINED 384 CHILDREN AND FOUND THAT 46 OF THEM SHOWED SIGNS OF AN ABNORMALITY. IS THIS STRONG EVIDENCE THAT THE RISK HAS INCREASED? ( WE CONSIDER A P – VALUE OF AROUND 5% TO REPRESENT STRONG EVIDENCE.)

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EXAMPLE 2 CONT’D

• (A) WRITE APPROPRIATE HYPOTHESES.• (B) CHECK THE NECESSARY ASSUMPTIONS.

• (C) PERFORM THE MECHANICS OF THE TEST. WHAT IS THE P – VALUE?

• (D) EXPLAIN CAREFULLY WHAT THE P – VALUE MEANS IN THIS CONTEXT.

• (E) WHAT’S YOUR CONCLUSION?• (F) DO ENVIRONMENTAL CHEMICALS CAUSE

CONGENITAL ABNORMALITIES?

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SOLUTION

INFERENCES ABOUT MEANS

• TESTING HYPOTHESES ABOUT MEANS

• ONE – SAMPLE t – TEST FOR MEANS

• PROBLEM• Test HO: = 0

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ASSUMPTIONS AND CONDITIONS

• INDEPENDENCE ASSUMPTION

• RANDOMIZATION CONDITION

• 10% CONDITION

• NEARLY NORMAL CONDITION OR LARGE SAMPLE

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STEPS IN TESTING

• NULL HYPOTHESIS

• HO: = 0

• ALTERNATIVE HYPOTHESIS

• HA: > 0

• or HA: < 0

• or HA: ≠ 0

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ATTITUDE: Assume that the null hypothesis HO is true and uphold it, unless data strongly speaks against it.

• STANDARD ERROR

• TEST STATISTICS

• t HAS STUDENT’S t – DISTRIBUTION WITH

n – 1 DEGREES OF FREEDOM.

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 P-value: Let to be the observed value of the

test statistic.

HA P-value NORMAL DISTRIBUTION CURVE 

HA: > 0 P(t > to)  

HA: < 0 P(t <to)  

HA: ≠ 0 P(t > |to|) + P(t < -|to|)

 

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CONCLUSION

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EXAMPLES FROM PRACTICE SHEET

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