Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Fall 2016 Room 150 Harvill Building 10:00 - 10:50 Mondays, Wednesdays & Fridays. Welcome http://today.msnbc.msn.com/id/33411196/ns/today-today_health/ http://www.youtube.com/watch?v=0r7NXEWpheg

By the end of lecture today 10/26/16 Introduction to Hypothesis Testing Type I versus Type II Errors

Before next exam (November 18th) Please read chapters 1 - 11 in OpenStax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

No Homework Due Friday 10/28/16

Lab sessions Everyone will want to be enrolled in one of the lab sessions Labs continue this week With Project 3

99% 95% 90% Moving from descriptive stats into inferential stats…. Area outside confidence interval is alpha Area outside confidence interval is alpha Moving from descriptive stats into inferential stats…. 99% Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there 95% Measurements that occur outside this middle ranges are suspicious, may be an error or belong elsewhere 90%

Confidence Interval of 95% Has and alpha of 5% α = .05 Critical z -2.58 Critical z 2.58 Confidence Interval of 99% Has and alpha of 1% α = .01 99% Critical z separates rare from common scores Critical z -1.96 Critical z 1.96 Confidence Interval of 95% Has and alpha of 5% α = .05 95% Area associated with most extreme scores is called alpha Critical z -1.64 Critical z 1.64 Confidence Interval of 90% Has and alpha of 10% α = . 10 90% Area in the tails is called alpha

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 don’t 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 That’s why we care about the z scores that define the middle 95% of the curve

.. 95% 95% X X Reject the null hypothesis 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 notnull big z score x x 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 then we have support for our alternative hypothesis Alternative 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 x x small z score

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? (α = .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 95% Has and alpha of 5% α = .05 Critical z -2.58 Critical z 2.58 Confidence Interval of 99% Has and alpha of 1% α = .01 99% Area in the tails is called alpha Critical z -1.96 Critical z 1.96 Confidence Interval of 95% Has and alpha of 5% α = .05 95% Critical Z separates rare from common scores Critical z -1.64 Critical z 1.64 Confidence Interval of 90% Has and alpha of 10% α = . 10 90% It would be easiest to reject the null at which alpha level? why?

Deciding whether or not to reject the null hypothesis. 05 versus 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 Significance level = .05 α = 0.01 Significance level = .01 -1.96 or +1.96 p < 0.05 Yes, Significant difference Reject the null Remember, reject the null if the observed z is bigger than the critical z -2.58 or +2.58 Not a Significant difference Do not Reject the null

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 x2) 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

Deciding whether or not to reject the null hypothesis. 05 versus 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 Significance level = .05 α = 0.01 Significance level = .01 -1.96 or +1.96 Do Not Reject the null Not a Significant difference Remember, reject the null if the observed z is bigger than the critical z -2.58 or +2.58 Not a Significant difference Do Not Reject the null

Deciding whether or not to reject the null hypothesis. 05 versus 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 Significance level = .05 α = 0.01 Significance level = .01 -1.96 or +1.96 p < 0.05 Yes, Significant difference Reject the null Remember, reject the null if the observed z is bigger than the critical z -2.58 or +2.58 p < 0.01 Yes, Significant difference Reject the null

Deciding whether or not to reject the null hypothesis. 05 versus 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 Significance level = .05 α = 0.01 Significance level = .01 -1.96 or +1.96 p < 0.05 Yes, Significant difference Reject the null Remember, reject the null if the observed z is bigger than the critical z -2.58 or +2.58 Not a Significant difference Do not Reject the null

Rejecting the null hypothesis 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 then we have support for our alternative 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 hypothesis

Setting our decision threshold Area in the tails is alpha 99% α = .01 95% α = .05 90% α = .10 Setting our decision threshold Level of significance is called alpha (α) 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

How would the critical z change? 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% 2.5% How would the critical z change? Pros and cons…

One versus two tail test of significance 5% versus 1% alpha levels How would the critical z change? One-tailed Two-tailed α = 0.05 Significance level = .05 α = 0.01 Significance level = .01 1% 5% 2.5% .5% .5% 2.5% -1.64 or +1.64 -1.96 or +1.96 -2.33 or +2.33 -2.58 or +2.58

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 Two-tailed α = 0.05 Significance level = .05 α = 0.01 Significance level = .01 -1.64 or +1.64 -1.96 or +1.96 Remember, reject the null if the observed z is bigger than the critical z Reject the null Reject the null -2.33 or +2.33 -2.58 or +2.58 Do not Reject the null Do not Reject 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 Two-tailed α = 0.05 Significance level = .05 α = 0.01 Significance level = .01 -1.64 or +1.64 -1.96 or +1.96 Remember, reject the null if the observed z is bigger than the critical z Do not Reject the null Reject the null -2.33 or +2.33 -2.58 or +2.58 Do not Reject the null Do not Reject 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 Two-tailed α = 0.05 Significance level = .05 α = 0.01 Significance level = .01 -1.64 or +1.64 -1.96 or +1.96 Remember, reject the null if the observed z is bigger than the critical z Reject the null Reject the null -2.33 or +2.33 -2.58 or +2.58 Reject the null Do not Reject the null

Thank you! See you next time!!