Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2016 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays Welcome

Schedule of readings Study guide for Exam 4 is online Before our fourth and final exam (May 2nd) OpenStax Chapters 1 – 13 (Chapter 12 is emphasized) Plous Chapter 17: Social Influences Chapter 18: Group Judgments and Decisions

Homework On class website: Please complete homework worksheet #26 Multiple Regressions Worksheet Due: Wednesday, April 27th

By the end of lecture today 4/25/16 Multiple Regression Using multiple predictor variables (independent) to make predictions about a single predicted variable (dependent) Multiple regression coefficients (b) One regression coefficient for each independent variable

Lab sessions No more labs this semester

Sample memorandum and general instructions for Project 4 are both online Due this week in class

Review 50% is explained so the other 50% has yet to be explained (0.71 > 0.632) Review

Summary Intercept: suggests that we can assume each salesperson will sell at least 20.526 systems Slope: as sales calls increase by one, 11.579 more systems should be sold Review

Some useful terms 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) Coefficient of regression is name for “b” Residual is found by y – y’

Homework Review

For each additional hour worked, weekly pay will increase by $6.09 +0.92 positive strong The relationship between the hours worked and weekly pay is a strong positive correlation. This correlation is significant, r(3) = 0.92; p < 0.05 up down 55.286 6.0857 y' = 6.0857x + 55.286 207.43 85.71 .846231 or 84% 84% of the total variance of “weekly pay” is accounted for by “hours worked” For each additional hour worked, weekly pay will increase by $6.09

400 380 360 Wait Time 340 320 300 280 4 5 6 7 8 Number of Operators

No we do not reject the null Critical r = 0.878 No we do not reject the null -.73 negative strong The relationship between wait time and number of operators working is negative and moderate. This correlation is not significant, r(3) = 0.73; n.s. number of operators increase, wait time decreases 458 -18.5 y' = -18.5x + 458 365 seconds 328 seconds .53695 or 54% The proportion of total variance of wait time accounted for by number of operators is 54%. For each additional operator added, wait time will decrease by 18.5 seconds

39 36 33 30 27 24 21 Percent of BAs 45 48 51 54 57 60 63 66 Median Income

Percent of residents with a BA degree Critical r = 0.632 Yes we reject the null Percent of residents with a BA degree 10 8 0.8875 positive strong The relationship between median income and percent of residents with BA degree is strong and positive. This correlation is significant, r(8) = 0.89; p < 0.05. median income goes up so does percent of residents who have a BA degree 3.1819 0.0005 y' = 0.0005x + 3.1819 25% of residents 35% of residents .78766 or 78% The proportion of total variance of % of BAs accounted for by median income is 78%. For each additional $1 in income, percent of BAs increases by .0005

30 27 24 21 18 15 12 Crime Rate 45 48 51 54 57 60 63 66 Median Income

No we do not reject the null Critical r = 0.632 No we do not reject the null Crime Rate 10 8 -0.6293 negative moderate The relationship between crime rate and median income is negative and moderate. This correlation is not significant, r(8) = -0.63; p < n.s. [0.6293 is not bigger than critical of 0.632] . median income goes up, crime rate tends to go down 4662.5 -0.0499 y' = -0.0499x + 4662.5 2,417 thefts 1,418.5 thefts .396 or 40% The proportion of total variance of thefts accounted for by median income is 40%. For each additional $1 in income, thefts go down by .0499

Multiple regression equations Can use variables to predict behavior of stock market probability of accident amount of pollution in a particular well quality of a wine for a particular year which candidates will make best workers

Can use variables to predict which candidates will make best workers Measured current workers – the best workers tend to have highest “success scores”. (Success scores range from 1 – 1,000) Try to predict which applicants will have the highest success score. We have found that these variables predict success: Age (X1) Niceness (X2) Harshness (X3) Both 10 point scales Niceness (10 = really nice) Harshness (10 = really harsh) According to your research, age has only a small effect on success, while workers’ attitude has a big effect. Turns out, the best workers have high “niceness” scores and low “harshness” scores. Your results are summarized by this regression formula: Y’ = b1X 1+ b2X 2+ b3X 3 + a Y’ = b1 X1 + b2 X2 + b3 X 3 + a Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

According to your research, age has only a small effect on success, while workers’ attitude has a big effect. Turns out, the best workers have high “niceness” scores and low “harshness” scores. Your results are summarized by this regression formula: Y’ = b1 X1 + b2 X2 + b3 X 3 + a Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

According to your research, age has only a small effect on success, while workers’ attitude has a big effect. Turns out, the best workers have high “niceness” scores and low “harshness” scores. Your results are summarized by this regression formula: Y’ = b1 X1 + b2 X2 + b3 X 3 + a Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 Y’ = b1X 1 + b2X 2 + b3X 3+ a Y’ is the dependent variable “Success score” is your dependent variable. X1 X2 and X3 are the independent variables “Age”, “Niceness” and “Harshness” are the independent variables. Each “b” is called a regression coefficient. Each “b” shows the change in Y for each unit change in its own X (holding the other independent variables constant). a is the Y-intercept

Y’ = b1X 1 + b2X 2 + b3X 3+ a The Multiple Regression Equation – Interpreting the Regression Coefficients Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 b1 = The regression coefficient for age (X1) is “1” The coefficient is positive and suggests a positive correlation between age and success. As the age increases the success score increases. The numeric value of the regression coefficient provides more information. If age increases by 1 year and hold the other two independent variables constant, we can predict a 1 point increase in the success score.

Y’ = b1X 1 + b2X 2 + b3X 3+ a The Multiple Regression Equation – Interpreting the Regression Coefficients Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 b2 = The regression coefficient for age (X2) is “20” The coefficient is positive and suggests a positive correlation between niceness and success. As the niceness increases the success score increases. The numeric value of the regression coefficient provides more information. If the “niceness score” increases by one, and hold the other two independent variables constant, we can predict a 20 point increase in the success score.

Y’ = b1X 1 + b2X 2 + b3X 3+ a The Multiple Regression Equation – Interpreting the Regression Coefficients Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 b3 = The regression coefficient for age (X3) is “-75” The coefficient is negative and suggests a negative correlation between harshness and success. As the harshness increases the success score decreases. The numeric value of the regression coefficient provides more information. If the “harshness score” increases by one, and hold the other two independent variables constant, we can predict a 75 point decrease in the success score.

Victoria will have a Success Index of 740 Here comes Victoria, her scores are as follows: Prediction line: Y’ = b1X 1+ b2X 2+ b3X 3+ a Y’ = 1X 1+ 20X 2 - 75X 3 + 700 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 Age = 30 Niceness = 8 Harshness = 2 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 What would we predict her “success index” to be? Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 We predict Victoria will have a Success Index of 740 Y’ = (1)(30) + (20)(8) - 75(2) + 700 = 3.812 Y’ = 740 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

What would we predict her “success index” to be? Here comes Victoria, her scores are as follows: Prediction line: Y’ = b1X 1+ b2X 2+ b3X 3+ a Y’ = 1X 1+ 20X 2 - 75X 3 + 700 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 Age = 30 Niceness = 8 Harshness = 2 Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 What would we predict her “success index” to be? Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 Y’ = (1)(30) + (20)(8) - 75(2) + 700 We predict Victoria will have a Success Index of 740 Y’ = 740 = 3.812 Here comes Victor, his scores are as follows: Age = 35 Niceness = 2 Harshness = 8 We predict Victor will have a Success Index of 175 What would we predict his “success index” to be? Y' = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700 Y’ = (1)(35) + (20)(2) - 75(8) + 700 Y’ = 175

Can use variables to predict which candidates will make best workers We predict Victor will have a Success Index of 175 We predict Victoria will have a Success Index of 740 Who will we hire?

Conducting multiple regression analyses that are relevant and useful starts with measurement designed to decrease uncertainty “Anything can be measured. If a thing can be observed in any way at all, it lends itself to some type of measurement method. No matter how “fuzzy” the measurement is, it’s still a measurement if it tells you more than you knew before.” Douglas Hubbard Author “How to Measure Anything: Finding the value of “Intangibles” in Business”

“A problem well stated is a problem half solved” “Anything can be measured. If a thing can be observed in any way at all, it lends itself to some type of measurement method. No matter how “fuzzy” the measurement is, it’s still a measurement if it tells you more than you knew before.” Douglas Hubbard Author “How to Measure Anything: Finding the value of “Intangibles” in Business” “A problem well stated is a problem half solved” Charles Kettering (1876 – 1958), American inventor, holder of 300 patents, including electrical ignition for automobiles How do we operationally define and measure constructs that we care about? “It is better to be approximately right, than to be precisely wrong.” - Warren Buffett Measurements don’t have to be precise to be useful

Thank you! See you next time!!