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15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 INTEGRATED LEARNING CENTER ILC 120 BNAD 276: Statistical Inference in Management Spring 2016 Welcome Green sheets Schedule of readings Before our next exam (March 22nd) OpenStax Chapters 1 11

Plous (10, 11, 12 & 14) Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness Homework On class website: Please print and complete homework #11 and #12 Due Thursday Hypothesis Testing and Confidence Intervals By the end of lecture today 3/1/16 Confidence Intervals Logic of hypothesis testing Steps for hypothesis testing Levels of significance (Levels of alpha) what does p < 0.05 mean? what does p < 0.01 mean? One-tail versus Two-tail test Type I versus Type II Errors Confidence Interval of 99% Has and alpha of 1% = Critical z separates rare from common scores = = Area in the tails is called

. 10 Critical z 1.96 Critical z -1.96 95% .05 Confidence Interval of 90% Has and alpha of 10% Critical z 2.58 99% .01 Confidence Interval of 95% Has and alpha of 5% Area associated with most extreme scores is called alpha Critical z -2.58 Critical z 1.64 Critical z -1.64 90% Moving from descriptive stats into inferential stats. Measurements that occur within the middle part of the

curve are ordinary (typical) and probably belong there Area outside confidence interval is alpha 99% 95% Measurements that occur outside this middle ranges are suspicious, may be an error or belong elsewhere 90% Hypothesis testing: How do we know if something is going on? How rare/weird is rare/weird enough? Every day examples about when is weird, weird enough to think something is going on? Handing in blue versus white test forms Psychic friend guesses 999 out of 1,000 coin tosses right Cancer clusters how many cases before investigation Weight gain treatment one group gained an average of 1 pound more than other groupwhat if 10? Why do we care about the z scores that define the middle 95% of the curve? Inferential Statistics Hypothesis testing with z scores allows us to make inferences

about whether the sample mean is consistent with the known population mean. Is the mean of my observed sample consistent with the known population mean or did it come from some other distribution? Why do we care about the z scores that define the middle 95% of the curve? If the z score falls outside the middle 95% of the curve, it must be from some other distribution Main assumption: We assume that weird, or unusual or rare things dont happen If a score falls out into the 5% range we conclude that it must be actually a common score but from some other distribution Thats why we care about the z scores that define the middle 95% of the curve . Main assumption: We assume that weird, or unusual or rare things dont happen If a score falls out into the tails (low probability) we conclude that it must be a common score from some other distribution Im not an outlier I just havent found my distribution yet . Reject the null hypothesis .. 95%

Relative to this distribution I am unusual maybe even an outlier X 95% X Relative to this distribution I am utterly typical Do not reject the null hypothesis Rejecting the null hypothesis null big z score x not. null 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 then we reject the null hypothesis Alternative then we have support for our alternative hypothesis Hypothesis If the observed z falls within the critical z in the distribution (curve): then we know it is a common score and is likely to be part of this distribution, we conclude it must be from this distribution then we do not reject the null hypothesis then we do not have support for our alternative null small z

x . Main assumption: We assume that weird, or unusual or rare things dont happen If a score falls out into the tails (low probability) we conclude that it must be a common score from some other distribution Im not an outlier I just havent found my distribution yet Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( = .05 or .01)? Critical z value? Step 3: Calculations from collected data observed z Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Step 5: Conclusion - tie findings back in to research problem Confidence Interval of 99% Has and alpha of 1% Area in the tails is called alpha = =

= It would be easiest to reject the null at which alpha why . 10 Critical z 1.96 Critical z -1.96 95% .05 Confidence Interval of 90% Has and alpha of 10% Critical z 2.58 99% .01 Confidence Interval of 95% Has and alpha of 5% Critical Z separates rare from common scores Critical z -2.58 Critical z 1.64 Critical z -1.64

90% 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? = 0.05 Remember, reject the null if the observed z is bigger than the critical z Significan ce level = .05 = 0.01 Significan ce level = .01 -1.96 or +1.96 p < 0.05 Reject the null -2.58 or Do not +2.58Reject the null Yes, Significant difference Not a Significant

difference Rejecting the null hypothesis e 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 b 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 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? = 0.05 Remember, reject the null if the observed z is bigger than the critical z Significan ce level = .05 = 0.01 Significan ce level = .01 -1.96 or +1.96 Do Not Reject the null

-2.58 or Do Not +2.58Reject the null Not a Significant difference Not a Significant difference 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? = 0.05 Remember, reject the null if the observed z is bigger than the critical z Significan ce level = .05 = 0.01 Significan ce level = .01 -1.96 or +1.96 p < 0.05 Reject the null

-2.58 or +2.58Reject the null Yes, Significant difference p < 0.01 Yes, Significant difference 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? = 0.05 Remember, reject the null if the observed z is bigger than the critical z Significan ce level = .05 = 0.01 Significan ce level = .01 -1.96 or +1.96 p < 0.05 Reject the null

-2.58 or Do not +2.58Reject the null Yes, Significant difference Not a Significant difference = .01 99% = .10 = .05 95% 90% Area in the tails is alpha Setting our decision threshold ( ) 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 One versus two tail test of significance:

Comparing different critical scores (but same alpha level e.g. alpha = 5%) One versus two tailed test of significance z score = 1.64 95% 95% 5% 2.5% How would the critical z change? Pros and cons 2.5% One versus two tail test of significance 5% versus 1% alpha levels How would the critical z change? One-tailed 5% 1% = 0.05 Significan ce level = .05 = 0.01 Significan ce level = .01 Two-tailed 2.5% .5% -1.64 or +1.64

-1.96 or +1.96 -2.33 or +2.33 -2.58 or +2.58 2.5% .5% One versus two tail test of significance 5% versus 1% alpha levels What if our observed z = 2.0? How would the critical z change? One-tailed = 0.05 Remember, reject the null if the observed z is bigger than the critical z Significan ce level = .05 -1.64 or Reject +1.64 the null -2.33 = 0.01 or

Do not Significan Reject+2.33 ce level the null = .01 Two-tailed -1.96 or +1.96 Reject the null -2.58 or Do not +2.58Reject the null One versus two tail test of significance 5% versus 1% alpha levels What if our observed z = 1.75? How would the critical z change? One-tailed = 0.05 Remember, reject the null if the observed z is bigger than the critical z Significan ce level = .05 -1.64 or Reject +1.64

the null -2.33 = 0.01 or Do not Significan Reject+2.33 ce level the null = .01 Two-tailed -1.96 or +1.96 Do not Reject the null -2.58 or Do not +2.58Reject the null One versus two tail test of significance 5% versus 1% alpha levels What if our observed z = 2.45? How would the critical z change? One-tailed = 0.05 Remember, reject the null if the observed z is bigger than the critical z Significan

ce level = .05 -1.64 or Reject +1.64 the null -2.33 = 0.01 or Reject +2.33 Significan the null ce level = .01 Two-tailed -1.96 or +1.96 Reject the null -2.58 or Do not +2.58Reject the null Logic of inferential stats. Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there (Do not reject the null hypothesis) Measurements that occur outside this middle range are

suspicious, may be an error or belong elsewhere (Do reject the null hypothesis) Area outside confidence interval is alpha 99% 95% 90% iew v Re Rejecting the null hypothesis e 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 b 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 iew v Re . 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 Do not reject Ho no snow globe move on Reject Ho yes snow globe, stop! Status of Null Hypothesis (actually, via magic truth-line) True Ho

False Ho No snow globe Yes snow globe You are right! Correct decision You are wrong! Type II error (miss) You are wrong! Type I error (false alarm) You are right! Correct decision Note: Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) Should we reject it???!! .Type True Ho I or type II error? Does this airline passenger Decision made have a snow globe? by experimenter Do not Reject Ho 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 she does have snow globe when she does have snow globe Say she doesnt have any when she doesnt have any Two ways to be incorrect: Say she does when she doesnt (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 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) Thank you! See you next time!!