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2. MGMT 276: Statistical Inference in Management Spring 2015
Welcome

4. Schedule of readings Before our fourth exam (April 30th) Lind
Chapter 13: Linear Regression and Correlation
Chapter 14: Multiple Regression
Chapter 15: Chi-Square
Plous
Chapter 17: Social Influences
Chapter 18: Group Judgments and Decisions

5. Over next couple of lectures 4/21/15
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)

6. Homework due – Thursday (April 23rd)
On class website:
Please print and complete homework worksheet #17
Multiple Regression Analyses

7. 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.

8. Do more sales calls result in more sales made?
Regression Example 60 70
Number of
sales calls made
systems sold 10 20 30 40 50
Ava
Emily
Do more sales calls result
in more sales made?
Isabella
Emma
Step 1: Draw scatterplot
Ethan
Step 2: Estimate r
Joshua
Jacob
Dependent
Variable
Independent
Variable

9. Regression Example
Do more sales calls result
in more sales made?
Step 3: Calculate r
Step 4: Is it a significant correlation?

10. Do more sales calls result in more sales made?
Step 4: Is it a significant correlation?
n = 10, df = 8
alpha = .05
Observed r is larger than critical r
(0.71 > 0.632)
therefore we reject the null hypothesis.
Yes it is a significant correlation
r (8) = 0.71; p < 0.05
Step 3: Calculate r
Step 4: Is it a significant correlation?

11. 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?

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

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

14. 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

15. 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

16. 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

17. 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

18. Regression: Evaluating Staff
Step 1: Compare expected sales levels to actual sales levels
Ava
Emma
Isabella
Emily
Madison
What should you expect from each salesperson
Joshua
Jacob
They should sell x systems depending on sales calls
If they sell more  over performing
If they sell fewer  underperforming

19. Regression: Evaluating Staff
Step 1: Compare expected sales levels to actual sales levels
=14.7
Difference between
expected Y’ and actual Y
is called “residual”
(it’s a deviation score)
Ava
14.7
How did
Ava do?
Ava sold 14.7 more than expected taking into account how many sales calls she made over performing

20. Regression: Evaluating Staff
Step 1: Compare expected sales levels to actual sales levels
=-23.7
Difference between
expected Y’ and actual Y
is called “residual”
(it’s a deviation score)
Ava
-23.7
How did
Jacob do?
Jacob sold fewer
than expected taking into account how many sales calls he
made under performing
Jacob

21. Regression: Evaluating Staff
Step 1: Compare expected sales levels to actual sales levels
Ava
Emma
Isabella
Emily
Madison
What should you expect from each salesperson
Joshua
Jacob
They should sell x systems depending on sales calls
If they sell more  over performing
If they sell fewer  underperforming

22. Regression: Evaluating Staff
Step 1: Compare expected sales levels to actual sales levels
Difference between
expected Y’ and actual Y
is called “residual”
(it’s a deviation score)
Ava
14.7
Emma
Isabella
-6.8
Emily
Madison
-23.7
7.9
Joshua
Jacob

23. 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?
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

24. Σ(Y – Y’) = 0 Σ(Y – Y’) Σx N Σ(Y – Y’)
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

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

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?
A note about curvilinear
relationships and patterns
of the residuals
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)

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

40. Homework Review

41. 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

42. 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 X b X b X a
Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

43. 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 X b X b X a
Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700

44. Y’ is the dependent variable
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 X b X b X a
Success score = (1)(Age) + (20)(Nice) + (-75)(Harsh) + 700
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

45. 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.

46. 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.

47. 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.

48. Thank you!
See you next time!!