1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays.
Welcome

2. A note on doodling

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

4. Homework on class website:
Please complete homework worksheets #25 and #26
Multiple Regression Worksheet and Test Review
Due: Wednesday, April 26th

5. Test review and tutoring
Lab sessions
Everyone will want to be enrolled
in one of the lab sessions
Optional
Test review and tutoring

6. By the end of lecture today 4/24/17
Multiple Regression
Using multiple predictor variables (independent) to make predictions about the predicted variable (dependent)
More than one coefficient of regression (also called “b”s or slopes)

7. Multiple Linear Regression - Example
Can we predict heating cost?
Three variables are thought to relate to the heating costs: (1) the mean daily outside temperature, (2) the number of inches of insulation in the attic, and (3) the age in years of the furnace.
To investigate, Salisbury's research department selected a random sample of 20 recently sold homes. It determined the cost to heat each home last January

13. Multiple Linear Regression - Example

14. The Multiple Regression Equation – Interpreting the Regression Coefficients
b1 = The regression coefficient for mean outside temperature
(X1) is
The coefficient is negative and shows a negative correlation between heating cost and temperature.
As the outside temperature increases, the cost to heat the home decreases. The numeric value of the regression coefficient provides more information. If we increase temperature by 1 degree and hold the other two independent variables constant, we can estimate a decrease of $4.583 in monthly heating cost.

15. The Multiple Regression Equation – Interpreting the Regression Coefficients
b2 = The regression coefficient for mean attic insulation (X2) is
The coefficient is negative and shows a negative correlation between heating cost and insulation.
The more insulation in the attic, the less the cost to heat the home.
So the negative sign for this coefficient is logical. For each additional inch of insulation, we expect the cost to heat the home to decline $14.83 per month, regardless of the outside temperature or the age of the furnace.

16. The Multiple Regression Equation – Interpreting the Regression Coefficients
b3 = The regression coefficient for mean attic insulation (X3) is 6.101
The coefficient is positive and shows a negative correlation between heating cost and insulation.
As the age of the furnace goes up, the cost to heat the home increases.
Specifically, for each additional year older the furnace is, we expect the cost to increase $6.10 per month.

18. Applying the Model for Estimation
What is the estimated heating cost for a home if:
the mean outside temperature is 30 degrees,
there are 5 inches of insulation in the attic, and
the furnace is 10 years old?

19. Preview Homework

20. r(18) = - 0.50 r(18) = - 0.40 r(18) = + 0.60 r(18) = - 0.811508835
500
400
300
200
100
500
400
300
200
100
500
400
300
200
100
Heating Cost
Heating Cost
Heating Cost
Average Temperature
Insulation
Age of Furnace
r(18) =
r(18) =
r(18) =
r(18) =
r(18) =
r(18) =

21. r(18) = - 0.50 r(18) = - 0.40 r(18) = + 0.60 r(18) = - 0.811508835
500
400
300
200
100
500
400
300
200
100
500
400
300
200
100
Heating Cost
Heating Cost
Heating Cost
Average Temperature
Insulation
Age of Furnace
r(18) =
r(18) =
r(18) =
r(18) =
r(18) =
r(18) =

22.
Y’ =
427.19 x1 x2 x3

23.
Y’ =
427.19 x1 x2 x3

24.
Y’ =
427.19 x1 x2 x3

25.
Y’ =
427.19 x1 x2 x3

26.
Y’ =
427.19 x1 x2 x3

27. 4.58
14.83
6.10
Y’ =
427.19
(30)
(5)
(10)
Y’ =
427.19
= $
= $
Calculate the predicted heating cost using the new value for the age of the furnace
Use the regression coefficient for the furnace ($6.10), to estimate the change

28. These differ by only one year but heating cost changed by $6.10
4.58
14.83
6.10
Y’ =
427.19
(30)
(5)
(10)
Y’ =
427.19
= $
Y’ =
427.19
(30)
(5)
(10)
These differ by only one year but heating cost changed by $6.10
– = 6.10
Y’ =
427.19
= $
= $ $
Y’ =
427.19
(30)
(5)
(11)
Y’ =
427.19
= $
Calculate the predicted heating cost using the new value for the age of the furnace
Use the regression coefficient for the furnace ($6.10), to estimate the change

29. 4.0
3.0
2.0
1.0
4.0
3.0
2.0
1.0
4.0
3.0
2.0
1.0
GPA
GPA
GPA
High School GPA
SAT (Verbal)
SAT (Mathematical)
r(7) = 0.50
r(7) =
r(7) =
r(7) =
r(7) =
r(7) =

30. 4.0
3.0
2.0
1.0
4.0
3.0
2.0
1.0
4.0
3.0
2.0
1.0
GPA
GPA
GPA
High School GPA
SAT (Verbal)
SAT (Mathematical)
r(7) = 0.50
r(7) =
r(7) =
r(7) =
r(7) =
r(7) =

31. 4.0
3.0
2.0
1.0
4.0
3.0
2.0
1.0
4.0
3.0
2.0
1.0
GPA
GPA
GPA
High School GPA
SAT (Verbal)
SAT (Mathematical)
r(7) = 0.50
r(7) =
r(7) =
r(7) =
r(7) =
r(7) =

32. 4.0
3.0
2.0
1.0
4.0
3.0
2.0
1.0
4.0
3.0
2.0
1.0
GPA
GPA
GPA
High School GPA
SAT (Verbal)
SAT (Mathematical)
r(7) = 0.50
r(7) =
r(7) =
r(7) =
r(7) =
r(7) =

33. No

34. No
Yes

35. No
Yes
0.0016 No

36. No
Yes
0.0016 No No

37. No
Yes
0.0016 No No
High School GPA

38. No
Yes
0.0016 No No
High School GPA
Y’ = x1 x2 x3

39. 1.201
.0016
.0019
Y’ = x1 x2 x3
Y’ =
(2.8)
(430)
(460)
= 2.76
2.76

40. 1.201
.0016
.0019
Y’ = x1 x2 x3
Y’ =
(3.8)
(430)
(460)
= 3.96
3.96

41. 1.201
.0016
.0019
2.76
3.96
= 1.2
Yes, use the regression coefficient for the HS GPA (1.2), to estimate the change

42. Let’s try one
When using hypothesis testing for correlation, what is our null hypothesis?
There is no relationship between the variables (r = 0)
There is a relationship between the variables (r ≠ 0)
Not enough info is given
Correct

43. Let’s try one
When using hypothesis testing for correlation, if we reject the null, what are we concluding?
There is no relationship between the variables (r = 0)
There is a relationship between the variables (r ≠ 0)
Not enough info is given
Correct

44. Let’s try one
Winnie found an observed correlation coefficient of 0, what should she conclude?
a. Reject the null hypothesis
b. Do not reject the null hypothesis
c. Not enough info is given
Correct

45. Y’ = a + bx1 + bx2 + bx3 + bx4 Correct
In the regression equation, what does the letter "a" represent?  a. Y intercept b. Slope of the line c. Any value of the independent variable that is selected d. None of these
Correct
Y’ = a + bx1 + bx2 + bx3 + bx4

46. Correct Assume the least squares equation is Y’ = 10 + 20X.
What does the value of 10 in the equation indicate?  a. Y intercept b. For each unit increased in Y, X increases by 10 c. For each unit increased in X, Y increases by 10 d. None of these .
Correct

47. In the least squares equation,  Y’  = X the value of 20 indicates  a. the Y intercept. b  slope (so for each unit increase in X, Y’ increases by 20). c. slope (so for each unit increase in Y’, X increases by 20). d. none of these.
Correct

48. In the equation Y’ = a + bX, what is Y’. a. Slope of the line b
In the equation  Y’  = a + bX, what is  Y’ ?  a. Slope of the line b. Y intercept C. Predicted value of Y, given a specific X value d. Value of Y when X = 0
Correct

49. Y’ = a + bx1 + bx2 + bx3 + bx4 Correct
If there are four independent variables in a multiple regression equation, there are also four  a. Y-intercepts (a). b. regression coefficients (slopes or bs). c. dependent variables. d. constant terms (k).
Correct
Y’ = a + bx1 + bx2 + bx3 + bx4

50. According to the Central Limit Theorem, which is false?
As n ↑ x will approach µ b.
As n ↑ curve will approach normal shape c.
As n ↑ curve variability gets larger
Correct
As n ↑ d.

51. Thank you!
See you next time!!