1. Tests of Significance about Percents Reading Handout # 6

2. Recall Example 1: SAT scores are falling? u 3 years ago -- National AVG = 955 u Random Sample of 200 graduating high school students this year : AVG = 935 SD = 100 Question: Have SAT scores dropped ? Procedure: Determine how “extreme” or “rare” our sample AVG of 935 is if population AVG really is 955.

3. u We found the P-value to be.22%

4. We must decide: u The sample came from population with population AVG = 955 and just by chance the sample AVG is “small.” OR u We are not willing to believe that the pop. AVG this year is really 955. (Conclude SAT scores have fallen.)

5. Hypothesis Testing Logic u Null u Null -- “nothing new is happening” u Alternative u Alternative -- what we “want” to show u Collect data u If - data supports alternative, meaning that “outcomes this extreme in support of the alternative could occur very rarely (< 5% of the time) when null is true” reject the null Then we reject the null.

6. SAT Example u We found the P-value to be.22% u So - results were highly statistically significant - i.e. we reject the null hypothesis that AVG is still 955 - we believe SAT scores have fallen

7. Example 2 - Last Time (school district data): - P-value = 11.5% - not rare enough - do not reject null - doesn’t mean we have proved AVG=100 (we simply do not have enough evidence to reject it.)

8. SO: u If we have very strong evidence against the null, we reject it  we believe the alternative is true u If we don’t have very strong evidence against the null, we do not reject it  we realize that the null might not be true  we do not claim the null is true

9. Analogy with Criminal Trial Defendant innocent u Null hypothesis: Defendant innocent Defendant guilty u Alternative hypothesis: Defendant guilty u Decision procedure: Evidence must show guilt beyond a reasonable doubt (before we find guilty) - want only a very small chance that jury finds guilty when person is actually innocent - so there may be fairly large chance jury finds not guilty when defendant is guilty innocent - thus, a finding of not guilty does not necessarily indicate jury believes defendant is innocent

10. New Drug - designed to reduce chance of catching common cold Without Drug - chance of catching cold during 1 year period = 40% To Test Drug: 150 subjects selected at random and given drug 35% of these catch cold during next 12 months Is there significant evidence that the drug is effective?

12. Suppose that for companies involved with Internet sales, the industry standard is that 90% of orders are mailed within 48 hours. - because of complaints from customers, a consumer group believes a particular company does not meet this standard Consumer Group conducted a survey - 180 orders selected randomly - 157 of these were mailed within 48 hours Is there significant evidence to support this belief?