Introduction to Statistics for the Social Sciences SBS
Introduction to Statistics for the Social Sciences SBS 200, COMM 200, GEOG 200, PA 200, POL 200, or SOC 200 Lecture Section 001, Fall 2015 Room 150 Harvill Building 10: 00 - 10: 50 Mondays, Wednesdays & Fridays. http: //courses. eller. arizona. edu/mgmt/delaney/d 15 s_database_weekone_screenshot. xlsx
By the end of lecture today 10/26/15 Hypothesis testing Doing everything right – but still being wrong Type I versus Type II Errors
Before next exam (November 20 th) Please read chapters 1 - 11 in Open. Stax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence
Homework Assignment Go to D 2 L - Click on “Interactive Online Homework Assignments” Complete Assignment 15: HW 15 -Hypothesis Testing, Type I versus Type II Errors Due: Wednesday, October 28 th
Confidence Interval of 99% Has and alpha of 1% Area outside confidence interval is alpha Area in the tails is called alpha α =. 10 Critical z 1. 96 Critical z -1. 96 95% α =. 05 Confidence Interval of 90% Has and alpha of 10% Area associated with most extreme scores is called alpha 99% α =. 01 Confidence Interval of 95% Has and alpha of 5% Critical z 2. 58 Critical z -2. 58 Critical z 1. 64 Critical z -1. 64 90% ew i v Re
Homework Worksheet: Confidence interval uses SEM
29. 2 80. 8 Homework Worksheet: Problem 1 Upper boundary raw score x = mean + (z)(standard deviation) x = 55 + (+ 2. 58)(10) x = 80. 8 Lower boundary raw score x = mean + (z)(standard deviation) x = 55 + (- 2. 58)(10) x = 29. 2 Standard deviation = 10 Mean = 55 2. 58 sd sd . 99 29. 2 ? 55 80. 8 ?
29. 2 80. 8 Homework Worksheet: Problem 1 Upper boundary raw score x = mean + (z)(standard error mean) 51. 3 58. 7 x = 55 + (+ 2. 58)(1. 42) x = 58. 7 Lower boundary raw score x = mean + (z)(standard error mean) x = 55 + (- 2. 58)(1. 42) x = 51. 3 Standard deviation = 10 Mean = 55 10 49 2. 58 sem . 99 51. 3 ? 55 58. 7 ? 1. 42
29. 2 80. 8 Homework Worksheet: Problem 5 51. 3 58. 7 10. 2 29. 8 16. 9 23. 1 8. 02 8. 6 7. 8 8. 6 9. 18 4. 09 13. 11 8. 02 9. 18 2. 67 14. 5 7. 8 9. 4
α =. 01 α =. 10 α =. 05 99% 95% 90% Area in the tails is alpha How do we know how rare is rare enough? (α ) Level of significance is called alpha • The degree of rarity required for an observed outcome to be “weird enough” to reject the null hypothesis • Which alpha level would be associated with most “weird” or rare scores? Critical z: A z score that separates common from rare outcomes and hence dictates whether the null hypothesis should be retained (same logic will hold for “critical t”) If the observed z falls beyond the critical z in the distribution (curve) then it is so rare, we conclude it must be from some other distribution ew i v Re
Rejecting the null hypothesis The result is “statistically significant” if: • the observed statistic is larger than the critical statistic (which can be a ‘z” or “t” or “r” or “F” or x 2) observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x 2) to be big • the p value is less than 0. 05 (which is our alpha) p < 0. 05 If we want to reject the null, we want our “p” to be small!! • we reject the null hypothesis • then we have support for our alternative hypothesis ew i v Re
Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = 2. 0? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 p < 0. 05 Reject the null -2. 58 or Do not +2. 58 Reject Yes, Significant difference Not a Significant difference the null ew i v Re
Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = 1. 5? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 Do Not Reject the null -2. 58 or Do Not +2. 58 Reject Not a Significant difference the null ew i v Re
Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = -3. 9? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 p < 0. 05 Reject the null -2. 58 or Reject +2. 58 the null Yes, Significant difference p < 0. 01 Yes, Significant difference ew i v Re
Deciding whether or not to reject the null hypothesis. 05 versus. 01 alpha levels What if our observed z = -2. 52? How would the critical z change? α = 0. 05 Remember, reject the null if the observed z is bigger than the critical z Significance level =. 05 α = 0. 01 Significance level =. 01 -1. 96 or +1. 96 p < 0. 05 Reject the null -2. 58 or Do not +2. 58 Reject Yes, Significant difference Not a Significant difference the null ew i v Re
Moving from descriptive stats Critical z into inferential stats…. -1. 64 Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there 5% Measurements that occur outside this middle ranges are suspicious, may be an error or belong elsewhere For scores that fall into the middle range, we do not Critical z reject the null 1. 64 90% 5% Critical For scores thatof fall What percent Values into regions of thethe distribution will rejection, fall in region weof reject the null rejection ew i v Re http: //today. msnbc. msn. com/id/33411196/ns/today_health/ http: //www. youtube. com/watch? v=0 r 7 NXEWphe g
Rejecting the null hypothesis The result is “statistically significant” if: • the observed statistic is larger than the critical statistic observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x 2) to be big!! • the p value is less than 0. 05 (which is our alpha) p < 0. 05 If we want to reject the null, we want our “p” to be small!! • we reject the null hypothesis • then we have support for our alternative hypothesis A note on decision making following procedure versus being right relative to the “TRUTH”
. Decision making: Procedures versus outcome Best guess versus “truth” What does it mean to be correct? Why do we say: • “innocent until proven guilty” • “not guilty” rather than “innocent” Is it possible we got a verdict wrong?
. We make decisions at Security Check Points .
. Type I or Type II error? . Does this airline passenger have a snow globe? Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) – Should we reject it? ? ? !! As detectives, do we accuse her of brandishing a snow globe?
. Does this airline passenger have a snow globe? Are we correct or have we made a Type I or Type II error? Decision made by experimenter Status of Null Hypothesis (actually, via magic truth-line) True Ho No snow globe False Ho Yes snow globe You are right! “no snow globe move on” Correct decision You are wrong! Type II error (miss) You are wrong! Type I error (false alarm) You are right! Correct decision Do not reject Ho Reject Ho “yes snow globe, stop!” Note: Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) – Should we reject it? ? ? !!
. Type I or type II error? Does this airline passenger Decision made have a snow globe? by experimenter True Ho Do not Reject Ho False Ho You are right! Correct decision You are wrong! Type II error (miss) You are wrong! You are right! Type I error Correct Reject Ho (false alarm) decision Two ways to be correct: • Say she does have snow globe when she does have snow globe • Say she doesn’t have any when she doesn’t have any Two ways to be incorrect: • Say she does when she doesn’t (false alarm) • Say she does not have any when she does (miss) What would null hypothesis be? This passenger does not have any snow globe Which is worse? Type I error: Rejecting a true null hypothesis Saying the she does have snow globe when in fact she does not (false alarm) Type II error: Not rejecting a false null hypothesis Saying she does not have snow globe when in fact she does (miss)
True Ho . Type I or type II error Does advertising affect sales? Do not Reject Ho Decision made by experimenter You are right! Correct decision False Ho You are wrong! Type II error (miss) You are wrong! You are right! Type I error Correct Reject Ho (false alarm) decision Two ways to be correct: • Say it helps when it does • Say it does not help when it doesn’t help Two ways to be incorrect: • Say it helps when it doesn’t • Say it does not help when it does Which is worse? What would null hypothesis be? This new advertising has no effect on sales Type I error: Rejecting a true null hypothesis Saying the advertising would help sales, when it really wouldn’t help people (false alarm) Type II error: Not rejecting a false null hypothesis Saying the advertising would not help when in fact it would (miss)
. What is worse a type I or type II error? What if we were looking at a new HIV drug that had no unpleasant side affects Do not Reject Ho Decision made by experimenter True Ho False Ho You are right! Correct decision You are wrong! Type II error (miss) You are wrong! You are right! Type I error Correct Reject Ho (false alarm) decision Two ways to be correct: • Say it helps when it does • Say it does not help when it doesn’t help Two ways to be incorrect: • Say it helps when it doesn’t • Say it does not help when it does Which is worse? What would null hypothesis be? This new drug has no effect on HIV Type I error: Rejecting a true null hypothesis Saying the drug would help people, when it really wouldn’t help people (false alarm) Type II error: Not rejecting a false null hypothesis Saying the drug would not help when in fact it would (miss)
. Type I or type II error Which is What if we were looking to see if there is a fire worse? burning in an apartment building full of cute puppies Two ways to be correct: • Say “fire” when it’s really there • Say “no fire” when there isn’t one Two ways to be incorrect: • Say “fire” when there’s no fire (false alarm) • Say “no fire” when there is one (miss) What would null hypothesis be? No fire is occurring Type I error: Rejecting a true null hypothesis (false alarm) Type II error: Not rejecting a false null hypothesis (miss)
. Type I or type II error What if we were looking to see if an individual were guilty of a crime? Which is worse? Two ways to be correct: • Say they are guilty when they are guilty • Say they are not guilty when they are innocent Two ways to be incorrect: • Say they are guilty when they are not • Say they are not guilty when they are What would null hypothesis be? This person is innocent - there is no crime here Type I error: Rejecting a true null hypothesis Saying the person is guilty when they are not (false alarm) Sending an innocent person to jail (& guilty person to stays free) Type II error: Not rejecting a false null hypothesis Saying the person in innocent when they are guilty (miss) Allowing a guilty person to stay free
. The null hypothesis is typically that something is not present, that there is no effect, that there is no difference between population and sample or between treatment and control. Null Hypothesis. A measure of sickness people taking drug people not taking drug people taking drug measureofof A measure sickness Drug does have effect people not taking drug (There are two distributions here, they are just on top of each other) (overlapping) Null is FALSE Something going on There is a difference between the groups Null is TRUE No effect of drug Nothing going on There is no difference between the groups
. (There are two distributions here, they are just on top of each other) (overlapping) A measure of sickness Remember: “procedure” vs “TRUTH” people taking drug A measureofof sickness Drug does have effect people taking drug people not taking drug Null is FALSE critical stat Score should fall in one of these regions people not taking drug Null is TRUE critical stat Score should fall in one of these regions Null is FALSE Something going on Score should fall in this critical stat region Null is TRUE No effect of drug Nothing going on
- Slides: 31