1. 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

3. Homework
On class website:
No Homework Due: Wednesday, April 20th

4. By the end of lecture today 4/18/16
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)

5. Schedule of readings 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

12. Labs will meet this week
Lab sessions
Labs will meet this week
Project 4

13. Review Regression Example
Rory is an owner of a small software company and employs 10 sales staff. Rory send his staff all over the world consulting, selling and setting up his system. He wants to evaluate his staff in terms of who are the most (and least) productive sales people and also whether more sales calls actually result in more systems being sold. So, he simply measures the number of sales calls made by each sales person and how many systems they successfully sold.
Review

14. Regression: Predicting sales
Step 1: Draw prediction line
r = 0.71
b = (slope)
a = (intercept)
Draw a regression line
and regression equation
What are we predicting?
Review

15. Regression: Predicting sales
Step 1: Draw prediction line
r = 0.71
b = (slope)
a = (intercept)
Draw a regression line
and regression equation
Review

16. Regression: Predicting sales
Step 1: Draw prediction line
r = 0.71
b = (slope)
a = (intercept)
Draw a regression line
and regression equation
Review

17. Rory’s Regression: Predicting sales from number of visits (sales calls)
Describe relationship
Regression line (and equation)
r = 0.71
Correlation: This is a strong positive correlation. Sales tend to increase as sales calls increase
Predict using regression line
(and regression equation)
b = (slope)
Slope: as sales calls increase by 1, sales should increase by
Dependent
Variable
Intercept: suggests that we can assume each salesperson will sell at least systems
a = (intercept)
Independent
Variable
Review

18. Regression: Predicting sales
You should sell systems
Step 1: Predict sales for a certain number of sales calls
Madison
Step 2: State the regression equation
Y’ = a + bx
Y’ = x
Joshua
If make one sales call
Step 3: Solve for some value of Y’
Y’ = (1)
Y’ =
What should you expect from a salesperson who makes 1 calls?
They should sell systems
If they sell more  over performing
If they sell fewer  underperforming
Review

19. Regression: Predicting sales
You should sell systems
Step 1: Predict sales for a certain number of sales calls
Isabella
Step 2: State the regression equation
Y’ = a + bx
Y’ = x
Jacob
If make two sales call
Step 3: Solve for some value of Y’
Y’ = (2)
Y’ =
What should you expect from a salesperson who makes 2 calls?
They should sell systems
If they sell more  over performing
If they sell fewer  underperforming
Review

20. Regression: Predicting sales
You should sell systems
Ava
Step 1: Predict sales for a certain number of sales calls
Emma
Step 2: State the regression equation
Y’ = a + bx
Y’ = x
If make three
sales call
Step 3: Solve for some value of Y’
Y’ = (3)
Y’ =
What should you expect from a salesperson who makes 3 calls?
They should sell systems
If they sell more  over performing
If they sell fewer  underperforming
Review

21. Regression: Predicting sales
You should sell systems
Step 1: Predict sales for a certain number of sales calls
Emily
Step 2: State the regression equation
Y’ = a + bx
Y’ = x
If make four sales calls
Step 3: Solve for some value of Y’
Y’ = (4)
Y’ =
What should you expect from a salesperson who makes 4 calls?
They should sell systems
If they sell more  over performing
If they sell fewer  underperforming
Review

22. Does the prediction line perfectly the predicted variable when using the predictor variable?
No, we are wrong sometimes…
How can we estimate how much “error” we have?
Exactly?
Difference between
expected Y’ and actual Y
is called “residual”
(it’s a deviation score)
14.7
How would we find our “average residual”?
-23.7
The green lines show how
much “error” there is in our
prediction line…how much
we are wrong in our predictions
Review

23. Σ(Y – Y’) = 0 Σ(Y – Y’) Σx N Σ(Y – Y’) Review
Residual scores
How do we find the average amount of error in our prediction
Ava is 14.7
Jacob is -23.7
Emily is -6.8
Madison is 7.9
The average amount by which actual scores
deviate on either side of the predicted score
Step 1: Find error for each value
(just the residuals)
Y – Y’
Difference between
expected Y’ and actual Y
is called “residual”
(it’s a deviation score)
Step 2: Add up the residuals
Big problem
Σ(Y – Y’) = 0
Square the deviations
Σ(Y – Y’) 2
How would we find our “average residual”? N Σx
Square root 2
n - 2
Σ(Y – Y’)
The green lines show how
much “error” there is in our
prediction line…how much
we are wrong in our predictions
Divide by df
Review

24. How do we find the average amount of error in our prediction
Deviation scores
Diallo is 0”
Preston is 2”
Mike is -4”
Step 1: Find error for each value
(just the residuals)
Hunter is -2
Y – Y’
Sound familiar??
Step 2: Find average √
∑(Y – Y’)2
Difference between
expected Y’ and actual Y
is called “residual”
(it’s a deviation score)
n - 2
How would we find our “average residual”? N Σx
The green lines show how
much “error” there is in our
prediction line…how much
we are wrong in our predictions
Review

25. These would be helpful to know by heart – please memorize
Standard error
of the estimate (line) =
These would be helpful to know by heart – please memorize
these formula
Review

26. Standard error of the estimate:
How well does the prediction line predict the predicted variable when using the predictor variable?
Standard error
of the estimate (line)
What if we want to know the “average deviation score”? Finding the standard error of the estimate (line)
Standard error of the estimate:
a measure of the average amount of predictive error
the average amount that Y’ scores differ from Y scores
a mean of the lengths of the green lines
Slope doesn’t give “variability” info
Intercept doesn’t give “variability” info
Correlation “r” does give “variability” info
Residuals do give “variability” info

27. How well does the prediction line predict the Ys from the Xs?
Residuals
Shorter green lines suggest better prediction – smaller error
Longer green lines suggest worse prediction – larger error
Why are green lines vertical?
Remember, we are predicting the variable on the Y axis
So, error would be how we are wrong about Y (vertical)

28. No, we are wrong sometimes…
Does the prediction line perfectly the predicted variable when using the predictor variable?
No, we are wrong sometimes…
How can we estimate how much “error” we have?
14.7
Difference between
expected Y’ and actual Y
is called “residual”
(it’s a deviation score)
-23.7
The green lines show how
much “error” there is in our
prediction line…how much
we are wrong in our predictions
Perfect correlation = or -1.00
Each variable perfectly
predicts the other
No variability in the scatterplot
The dots approximate a straight line

29. Regression Analysis – Least Squares Principle
When we calculate the regression line we try to:
minimize distance between predicted Ys and actual (data) Y points (length of green lines)
remember because of the negative and positive values cancelling each other out we have to square those distance (deviations)
so we are trying to minimize the “sum of squares of the vertical distances between the actual Y values and the predicted Y values”

30. Is the regression line better than just guessing the mean of the Y variable? How much does the information about the relationship actually help?
Which minimizes error better?
How much better does the regression line predict the observed results? r2
Wow!

31. r2 = The proportion of the total variance in one variable that is
What is r2?
r2 = The proportion of the total variance in one variable that is
predictable by its relationship with the other variable
Examples
If mother’s and daughter’s heights are
correlated with an r = .8, then what amount (proportion or percentage)
of variance of mother’s height is accounted for by daughter’s height?
.64 because (.8)2 = .64

32. r2 = The proportion of the total variance in one variable that is
What is r2?
r2 = The proportion of the total variance in one variable that is
predictable for its relationship with the other variable
Examples
If mother’s and daughter’s heights are
correlated with an r = .8, then what proportion of variance of mother’s height
is not accounted for by daughter’s height?
.36 because ( ) = .36 or
36% because 100% - 64% = 36%

33. If ice cream sales and temperature are correlated with an
What is r2?
r2 = The proportion of the total variance in one variable that is
predictable for its relationship with the other variable
Examples
If ice cream sales and temperature are correlated with an
r = .5, then what amount (proportion or percentage) of variance of ice cream sales is accounted for by temperature?
.25 because (.5)2 = .25

34. If ice cream sales and temperature are correlated with an
What is r2?
r2 = The proportion of the total variance in one variable that is
predictable for its relationship with the other variable
Examples
If ice cream sales and temperature are correlated with an
r = .5, then what amount (proportion or percentage) of variance of ice cream sales is not accounted for by temperature?
.75 because ( ) = .75 or
75% because 100% - 25% = 75%

35. 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)
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)

36. Thank you!
See you next time!!