Apr. 5

1. Apr. 5 Stat 100

2. To do • Read Chapter 21, try problems 1-6 • Skim Chapter 22

3. Thought Question • Suppose we think mean classes missed per week is higher for males than for females. • How would we determine whether this is the case?

4. Basic Strategy • Collect data on missed classes for males and females • Calculate mean for each sex and determine the difference. • Then what? Suppose the mean for 50 males was 0.5 higher than mean for 50 females. • Could we generalize that for all students, mean is higher for males?

5. An Issue • Did the observed difference occur just by chance or does it reflect a true difference? • What’s the likelihood the observed difference would be 0.5 if there’s really no difference in the populations?

6. Significance Test • Data used to decide between two competing statements (hypotheses) about the population • Statements are called null hypothesis and alternative hypothesis

7. The two hypotheses • null hypothesis : a “nothing happening” statement • no difference between groups, or no relationship, or nothing new • alternative hypothesis : "something's happening” • there is a difference, or there is a relationship, or something’s new

8. Notation • H0 represents null hypothesis • HA represents alternative hypothesis

9. Missed classes by men and women • H0: No difference in mean classes missed for men and women • HA: There is a difference between mean missed classes by men and women

10. Deciding between the hypotheses – an example • Students asked to randomly a number between 1 and 10 • In past, a bias toward picking 7 has been noticed • With random picking, what proportion would pick 7? • Answer = 1/10 = 0.1 because there are 10 numbers

11. Are students picking randomly? • let p represent proportion of all students would pick 7 • null : random picking , p = 0.10 • alternative: bias toward 7 , p> 0.10

12. The sample data • Suppose 24 of 100 students in the class pick 7. • The proportion picking 7 is 24/100 = 0.24 (much higher than .10) • Could this have happened through random picking? • Or, does it reflect bias toward 7?

13. With true random picking • Distribution of possible sample proportions would be bell curve • Centered at 0.10 (chance of randomly picking 7) • SD = Sqr root[(0.1)(1-0.1)/100] = 0.03 • About 99.7% chance that sample proportion would fall in range 0.10  3(.03), or .01 to .19

14. Where the observed statistic falls • Sample value of 0.24 is outside interval of what might normally occur with random picking • Reasonable to reject the hypothesis that picking is random

15. Where the observed statistic falls • Observed = 0.24 picked 7 • Find percentile rank of 0.24 in bell curve with mean 0.10 and SD = 0.03 • z score = (0.24-0.10) / 0.03 = 0.14 / 0.03= 4.67 • Page 137 table, proportion to left of z is around 0.995

16. Statistically Significant • A result is called statistically significant when a null hypothesis is rejected • In our example, the result was statistically significant

17. Thought Question • Imagine all the criminal trials in the United States • In each trial, what is the null hypothesis? • What is the alternative hypothesis? • Overall criminal trials what are the two types of mistakes that can b made?

18. Statistical Errors • Type 1: rejecting the null hypothesis when you should not (like a convicting innocent person) • Type 2: not rejecting the null hypothesis when you should (like failing to convict a guilty person)

19. Example • Experiment is done to see if new treatment for depression is better than old treatment • null hyp: new treatment not better • alternative hyp: new treatment is better • Type 1 error: deciding new treatment is better when it is not • Type 2 error: deciding new treatment is not better when it actually is