Please hand in Project 4 To your TA

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

It went really well! Exam 3 – This past Friday Thanks for your patience and cooperation We should have the grades up by Friday (takes about a week)

Lab sessions Everyone will want to be enrolled in one of the lab sessions No Labs this week

Schedule of readings Before our fourth and final exam (December 5th) OpenStax Chapters 1 – 13 (Chapter 12 is emphasized) Plous Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions

By the end of lecture today 11/21/16 Please hand in Project 4 Logic of hypothesis testing with Correlations Interpreting the Correlations and scatterplots Simple Regression Using correlation for predictions

Homework On class website: Please complete homework worksheet #22 Correlations Worksheet Due: Monday, November 28th

Over next couple of lectures 11/21/16 Logic of hypothesis testing with Correlations Interpreting the Correlations and scatterplots Simple and Multiple Regression Using correlation for predictions r versus r2 Regression uses the predictor variable (independent) to make predictions about the predicted variable (dependent) Coefficient of correlation is name for “r” Coefficient of determination is name for “r2” (remember it is always positive – no direction info) Standard error of the estimate is our measure of the variability of the dots around the regression line (average deviation of each data point from the regression line – like standard deviation) Coefficient of regression will “b” for each variable (like slope)

No class on Wednesday Happy Holiday!

We’ll call the correlations “r” Correlation: Measure of how two variables co-occur and also can be used for prediction Range between -1 and +1 The closer to zero the weaker the relationship and the worse the prediction Positive or negative Remember, We’ll call the correlations “r” Revisit this slide

Remember, Correlation = “r” Revisit this slide Positive correlation as values on one variable go up, so do values for other variable pairs of observations tend to occupy similar relative positions higher scores on one variable tend to co-occur with higher scores on the second variable lower scores on one variable tend to co-occur with lower scores on the second variable scatterplot shows clusters of point from lower left to upper right Revisit this slide

Negative correlation Remember, Correlation = “r” Revisit this slide as values on one variable go up, values for other variable go down pairs of observations tend to occupy dissimilar relative positions higher scores on one variable tend to co-occur with lower scores on the second variable lower scores on one variable tend to co-occur with higher scores on the second variable scatterplot shows clusters of point from upper left to lower right Revisit this slide

Zero correlation Revisit this slide as values on one variable go up, values for the other variable go... anywhere pairs of observations tend to occupy seemingly random relative positions scatterplot shows no apparent slope Revisit this slide

Correlation does not imply causation Is it possible that they are causally related? Yes, but the correlational analysis does not answer that question What if it’s a perfect correlation – isn’t that causal? No, it feels more compelling, but is neutral about causality Number of Birthdays Remember the birthday cakes! Number of Birthday Cakes Revisit this slide

Correlation - How do numerical values change? Revisit this slide

Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Both axes have real numbers listed Both axes and values are labeled Variable name is listed clearly Revisit this slide

Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Both axes have real numbers listed Both axes and values are labeled Variable name is listed clearly

Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Both axes have real numbers listed Both axes and values are labeled Variable name is listed clearly Revisit this slide

Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Both axes have real numbers listed Both axes and values are labeled Variable name is listed clearly Revisit this slide

Height of Daughters (inches) Height of Mothers (in) 48 52 56 60 64 68 72 76 48 52 5660 64 68 72 This shows the strong positive (r = +0.8) relationship between the heights of daughters (in inches) with heights of their mothers (in inches). Variable name is listed clearly Description includes: Both variables Strength (weak,moderate,strong) Direction (positive, negative) Estimated value (actual number) Both axes have real numbers listed Both axes and values are labeled Variable name is listed clearly Statistically significant p < 0.05 Reject the null hypothesis Revisit this slide

Finding a statistically significant correlation The result is “statistically significant” if: the observed correlation is larger than the critical correlation we want our r to be big if we want it to be significantly different from zero!! (either negative or positive but just far away from zero) the p value is less than 0.05 (which is our alpha) we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses For correlation null is that r = 0 (no relationship) Step 2: Decision rule Alpha level? (α = .05 or .01)? Critical statistic (e.g. critical r) value from table? Degrees of Freedom = (n – 2) Step 3: Calculations df = # pairs - 2 Step 4: Make decision whether or not to reject null hypothesis If observed r is bigger than critical r then reject null Step 5: Conclusion - tie findings back in to research problem

Five steps to hypothesis testing Problem 1 Is there a relationship between the: Price Square Feet We measured 150 homes recently sold

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Is there a relationship between the cost of a home and the size of the home Describe the null and alternative hypotheses null is that there is no relationship (r = 0.0) alternative is that there is a relationship (r ≠ 0.0) Step 2: Decision rule – find critical r (from table) Alpha level? (α = .05) Degrees of Freedom = (n – 2) 150 pairs – 2 = 148 pairs df = # pairs - 2

α = .05 Critical r value from table df = 148 pairs Critical value r(148) = 0.195 df = # pairs - 2

Five steps to hypothesis testing Step 3: Calculations

Five steps to hypothesis testing Step 3: Calculations

Five steps to hypothesis testing Step 3: Calculations r = 0.726965 Critical value r(148) = 0.195 Observed correlation r(148) = 0.726965 Step 4: Make decision whether or not to reject null hypothesis If observed r is bigger than critical r then reject null Yes we reject the null 0.727 > 0.195

These data suggest a strong positive correlation Conclusion: Yes we reject the null. The observed r is bigger than critical r (0.727 > 0.195) Yes, this is significantly different than zero – something going on These data suggest a strong positive correlation between home prices and home size. This correlation was large enough to reach significance, r(148) = 0.73; p < 0.05

Finding a statistically significant correlation The result is “statistically significant” if: the observed correlation is larger than the critical correlation we want our r to be big if we want it to be significantly different from zero!! (either negative or positive but just far away from zero) the p value is less than 0.05 (which is our alpha) we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis

Correlation matrices Correlation matrix: Table showing correlations for all possible pairs of variables Education Age IQ Income 0.38* 1.0** 0.41* 0.65** IQ Age Education Income 0.41* -0.02 0.52* 1.0** 1.0** 0.27* 0.65** 1.0** Remember, Correlation = “r” * p < 0.05 ** p < 0.01 Revisit this slide

Correlation matrices Correlation matrix: Table showing correlations for all possible pairs of variables Education Age IQ Income 0.41* 0.38* 0.65** IQ Age Education Income -0.02 0.52* 0.27* * p < 0.05 ** p < 0.01

Correlation matrices Variable names Make up any name that means something to you VARX = “Variable X” VARY = “Variable Y” VARZ = “Variable Z” Correlation of X with X Correlation of Y with Y Correlation of Z with Z

Correlation matrices Variable names Make up any name that Does this correlation reach statistical significance? Variable names Make up any name that means something to you VARX = “Variable X” VARY = “Variable Y” VARZ = “Variable Z” Correlation of X with Y Correlation of X with Y p value for correlation of X with Y p value for correlation of X with Y

Correlation matrices Variable names Make up any name that Does this correlation reach statistical significance? Variable names Make up any name that means something to you VARX = “Variable X” VARY = “Variable Y” VARZ = “Variable Z” Correlation of X with Z Correlation of X with Z p value for correlation of X with Z p value for correlation of X with Z

Correlation matrices Variable names Make up any name that Does this correlation reach statistical significance? Variable names Make up any name that means something to you VARX = “Variable X” VARY = “Variable Y” VARZ = “Variable Z” Correlation of Y with Z Correlation of Y with Z p value for correlation of Y with Z p value for correlation of Y with Z

Correlation matrices What do we care about?

Thank you! See you next time!!