1. STATISTICAL INFERENCES
CHAPTERS 12 AND 13
HYPOTHESIS TESTING FOR PROPORTIONS AND MEANS

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.”

3. TESTING HYPOTHESES NULL 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

4. IN OUR PROBLEM

5. 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.

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

9. CONSIDERING THE EXAMPLE AT THE BEGINNING:

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.

11. THE WAY THE ALTERNATIVE HYPOTHESIS IS WRITTEN IS HELPFUL IN COMPUTING THE P - VALUE
NORMAL CURVE

12. IN OUR EXAMPLE,
P – VALUE = P( z < - 2.4) =
INTERPRETATION: IF THE COIN IS FAIR, THEN THE PROBABILITY OF OBSERVING 38 OR FEWER HEADS IN 100 TOSSES IS

13. 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

15. 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?

16. 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.

17. SOLUTION

18. 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.)

19. 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?

20. SOLUTION

21. INFERENCES ABOUT MEANS
TESTING HYPOTHESES ABOUT MEANS
ONE – SAMPLE t – TEST FOR MEANS
PROBLEM
Test HO:  = 0

22. ASSUMPTIONS AND CONDITIONS
INDEPENDENCE ASSUMPTION
RANDOMIZATION CONDITION
10% CONDITION
NEARLY NORMAL CONDITION OR LARGE SAMPLE

23. STEPS IN TESTING NULL HYPOTHESIS HO:  = 0 ALTERNATIVE HYPOTHESIS
HA:  > 0
or HA:  < 0
or HA:  ≠ 0

24. t HAS STUDENT’S t – DISTRIBUTION WITH n – 1 DEGREES OF FREEDOM.
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.

25. 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|)

26. CONCLUSION

27. EXAMPLES FROM PRACTICE SHEET