1 Bivariate Regression Straight Lines ¾ Simple way to describe a relationship ¾ Remember the equation for a straight line? z y = mx + b ¾ What is m? What is b? ¾ How do you compute the equation? (x1,y1) (x2,y2) What if every point is not on the line? ¾ Straight line may be good description even if not all points are on the line Computing the line when points are scattered ¾ = a + bX ¾ Y-hat means predicted value of Y ¾ Computing the slope: ¾ b = 𝑋−𝑋 𝑌−𝑌 𝑋−𝑋 ¾ I ill ri e/r n, b no e al o consider variability in X and Y Computing the intercept ¾ a = - bX ¾ Need o pl g in al e of (X, ) ¾ Can e j an Y or X! z Line would be very different depending on which ones you chose ¾ Must have X and Y that we know are on the line z mean of X and mean of Y 2 Computing the intercept ¾ Regression line will always go through the mean of X and mean of Y ¾ A = 𝑌 - b𝑋 ¾ Le r it with our example from before X (# of kids) Y (hours of housework) 𝑋 𝑋 𝑌 𝑌 𝑋 𝑋 𝑌 𝑌 𝑋 𝑋 1 1 -1.75 -2.5 4.375 3.063 1 2 -1.75 -1.5 2.625 3.063 1 3 -1.75 -0.5 0.875 3.063 2 6 -0.75 2.5 -1.875 0.563 2 4 -0.75 0.5 -0.375 0.563 2 1 -0.75 -2.5 1.875 0.563 3 5 0.25 1.5 0.375 0.063 3 0 0.25 -3.5 -0.875 0.063 4 6 1.25 2.5 3.125 1.563 4 3 1.25 -0.5 -0.625 1.563 5 7 2.25 3.5 7.875 5.063 5 4 2.25 0.5 1.125 5.063 MX=2.75 MY=3.5 = 0 = 0 = 18.5 = 24.25 Computing the equation ¾ b = . . .76 ¾ a = 3.5 - .76(2.75) ¾ = 1.41 ¾ = 1.41 + .76X Interpreting the coefficients ¾ Slope z For a one unit increase in X, we predict a b unit increase in Y What does that mean for this study? ¾ Intercept z The predicted value of Y when X = 0 What does that mean for this study? Interpreting the coefficients ¾ Slope z For each additional child, we predict parents will do an additional .76 hours of housework per day ¾ Intercept z For a family with zero kids, we predict they will do 1.41 hours of housework per day Drawing the regression line ¾ Need to plot two points z 𝑋, 𝑌 z Y-intercept 1 Scatterplots and Correlation Correlation ¾ Useful tool to assess relationships ¾ Must have two variables measured on one set of people ¾ Correlation only measures strength of linear association Linear relationships are not perfect lines ¾ Variables have variability (duh) ¾ Relationships may be generally linear even if all points are not on the line Magnitude of r Not all relationships are linear 2 Properties of r ¾ X & Y must be quantitative z Interval or ratio ¾ I doe n ma e hich a iable i edic o and which is response z rxy = ryx Properties of r ¾ Correlation has no units z So r can be compared for different variables ¾ Value of r is always between -1 and +1 Computing r ¾ Consider deviations around mean of X & Y ¾ (X 𝑋) (Y 𝑌) Cross-Product ¾ To consider X & Y together, multiply their deviations ¾ (X 𝑋)(Y 𝑌) ¾ Sign will be positive or negative ¾ Sum of cross-pr

1. 1
Bivariate Regression
Straight Lines
¾ Simple way to describe a relationship
¾ Remember the equation for a straight line?
z y = mx + b
¾ What is m? What is b?
¾ How do you compute the equation?
(x1,y1)
(x2,y2)
What if every point is
not on the line?
¾ Straight line may be good description even
if not all points are on the line
Computing the line
when points are scattered
¾ = a + bX
¾ Y-hat means predicted value of Y
¾ Computing the slope:
¾ b = �−� �−�

2. �−�
¾ I ill ri e/r n, b no e al o
consider variability in X and Y
Computing the intercept
¾ a = - bX
¾ Need o pl g in al e of (X, )
¾ Can e j an Y or X!
z Line would be very different depending on
which ones you chose
¾ Must have X and Y that we know are on
the line
z mean of X and mean of Y
2
Computing the intercept
¾ Regression line will always go through the
mean of X and mean of Y
¾ A = � - b�
¾ Le r it with our example from before
X
(# of kids)
Y
(hours of

3. housework) � � � � � � � � � �
1 1 -1.75 -2.5 4.375 3.063
1 2 -1.75 -1.5 2.625 3.063
1 3 -1.75 -0.5 0.875 3.063
2 6 -0.75 2.5 -1.875 0.563
2 4 -0.75 0.5 -0.375 0.563
2 1 -0.75 -2.5 1.875 0.563
3 5 0.25 1.5 0.375 0.063
3 0 0.25 -3.5 -0.875 0.063
4 6 1.25 2.5 3.125 1.563
4 3 1.25 -0.5 -0.625 1.563
5 7 2.25 3.5 7.875 5.063
5 4 2.25 0.5 1.125 5.063
MX=2.75 MY=3.5 = 0 = 0 = 18.5 = 24.25
Computing the equation
¾ b = .
.
.76
¾ a = 3.5 - .76(2.75)

4. ¾ = 1.41
¾ = 1.41 + .76X
Interpreting the coefficients
¾ Slope
z For a one unit increase in X, we predict a b
unit increase in Y
What does that mean for this study?
¾ Intercept
z The predicted value of Y when X = 0
What does that mean for this study?
Interpreting the coefficients
¾ Slope
z For each additional child, we predict
parents will do an additional .76 hours of
housework per day
¾ Intercept
z For a family with zero kids, we predict they
will do 1.41 hours of housework per day
Drawing the regression line
¾ Need to plot two points
z �, �
z Y-intercept

5. 1
Scatterplots and
Correlation
Correlation
¾ Useful tool to assess relationships
¾ Must have two variables measured on one set of
people
¾ Correlation only measures strength of linear
association
Linear relationships are
not perfect lines
¾ Variables have variability (duh)
¾ Relationships may be generally linear
even if all points are not on the line
Magnitude of r Not all relationships are linear
2
Properties of r
¾ X & Y must be quantitative
z Interval or ratio

6. ¾ I doe n ma e hich a iable i edic o
and which is response
z rxy = ryx
Properties of r
¾ Correlation has no units
z So r can be compared for different variables
¾ Value of r is always between -1 and +1
Computing r
¾ Consider deviations around mean of X & Y
¾ (X �) (Y �)
Cross-Product
¾ To consider X & Y together, multiply their
deviations
¾ (X �)(Y �)
¾ Sign will be positive or negative
¾ Sum of cross-products is an indication of
overall relationship
¾ S (X �)(Y �)
¾ Mostly positive, sum = positive
¾ Mostly negative, sum = negative
¾ About equal pos & neg, m 0
¾ Sum of cross-products can get very big if
N is large

7. ¾ So le adj fo N
S (X �)(Y �)
�
¾ This is called the Covariance
3
Bill McDonald Speed Trap
Subject X (Age) Y (Speed) X- � Y- � (X �)(Y �)
1 18 48 -10 3 -30
2 24 51 -4 6 -24
3 43 41 15 -4 -60
4 34 39 6 -6 -36
5 21 46 -7 1 -7
Sum: 140 225 0 0 -157
Mean: 28 45
SD: 9.23 4.42
Just for this sample, not
estimating population
¾ Covariance
= -157/5 = 31.4
¾ What does that mean?
z Is it big?
z Is it small?
z What are the units?
¾ We need to get this on a scale that makes

8. sense!
¾ Le anda di e i
z Divide by standard deviation of X and Y
S (X �)(Y �)
�����
¾ When you standardize the covariance, you
ge
¾ ha o ge i r
¾ The units cancel out!
. .
= -.77
Cohen g ideline fo Corr
¾ These are for the behavioral sciences only
z ~.10 = small
z ~.30 = medium
z ~.50 = large
Interpret in words!
¾ Younger drivers tended to be driving faster
when pulled over.
¾ Watch wording!
z Do interpret direction
z Don e ca al lang age
Being o ng doe n CAUSE eo le o d i e fa

9. 4
r as a measure of effect size
¾ Variance in a variable can be partitioned
z Explained + unexplained = total
John Venn
r2 is proportion of
variance explained
¾ rage,speed = -.77
¾ That means -.772 = .59 (59%) of variance in
speed is explained by or predicted by age
z Not necessarily caused by age
¾ 1 - .59 = .41
¾ 41% of the variability in speed is NOT
explained by age
Factors that affect size of r
¾ Biased samples
z Restricted range
tends to make corr smaller
z extreme groups
e.g., only use people really high or low on X
Tends to make corr larger
Design Issues

10. ¾ N ho ld be e big ( 30)
z Small samples are too unstable
¾ Wide variability on X & Y
z No restriction of range
Factors that affect size of r
¾ Combined groups
z May mask relationship in each group
¾ r is very affected by outliers
Psychology 302, Winter 2020
Problem Set 2, due Wednesday, Feb 5 IN CLASS
Do all problems except #2 by hand and show your work
1. A researcher wants to study the relationship between
extraversion and amount
of social interaction. She administers a measure of extraversion
that ranges
from 1-20, where higher scores mean higher extraversion. She
then observes
the number of social interactions between participants in a 30
minute period in
the lab. The data are as follows:
Participant

11. Extraversion score (X)
Number of social
interactions (Y)
1 20 7
2 5 2
3 18 9
4 6 3
5 19 8
6 8 6
7 15 3
8 7 4
9 17 7
10 16 10
Mean: 13.10 5.90
SD: 5.59 2.62
a. Draw a scatterplot for the data
b. Calculate the correlation of X and Y using the formula I
provided in
class
c. What proportion of variance in number of social interactions
is
explained by extraversion score?
d. Fully describe this relationship in English (not stats-speak).
That
means say something about the direction and strength of the
relationship and what it means.

12. 2. Re-do problem #1 (a) and (b) using SPSS. Follow the
guidelines in your
handout to make an attractive scatterplot. Hand in your SPSS
output and circle
the values of r on the output.
3. Given the following set of paired scores for 5 subjects:
Sub ID: 1 2 3 4 5
X 6 8 4 8 7
Y 5 6 9 9 11
a. Construct a scatter plot for the data
b. Compute the value of the correlation coefficient
c. Add the following set of scores from a sixth subject to the
data: X =
24, Y = 26
d. Add the new point to your scatterplot from (a) (or make a
new plot)
e. Compute the correlation for the set of six paired scores
f. Explain why there is such a big difference between (b) and
(d).
4. As the value of a correlation approaches r1.0 (compared to a
correlation close to
zero), what does it indicate about the following:

13. a. The shape of the scatterplot
b. The variability of the Y scores at each value of X
c. The accuracy with which we can predict Y if X is known
5. A recent study reported a relationship between anger levels
(X) and blood
pressure (Y) in college-age participants. You want to see
whether this is true in
a sample of 12 friends. You use an anger measure that ranges
from 10-100
where higher scores mean more anger. Systolic blood pressure
(SBP) is
measured in millimeters of mercury, where higher values mean
higher blood
pressure. I did a lot of the legwork for you on this one. Do the
remaining
calculations by hand and show your work.
X Y � − � � − � � − � � − �
64 170 25.5 31.5 803.25
27 132 -11.5 -6.5 74.75
37 129 -1.5 -9.5 14.25
39 108 0.5 -30.5 -15.25
17 122 -21.5 -16.5 354.75
29 118 -9.5 -20.5 194.75
36 131 -2.5 -7.5 18.75
34 142 -4.5 3.5 -15.75
44 156 5.5 17.5 96.25
50 168 11.5 29.5 339.25
46 147 7.5 8.5 63.75

14. 39 139 0.5 0.5 0.25
6 = 38.5 138.5 0 0 1929
Sx = 11.49 SY = 18.41
a. Calculate the correlation of anger level (X) and SBP (Y)
using the formula I
provided in class
b. What proportion of variance in anger is explained by SBP?
c. Fully describe this relationship in English (not stats-speak).
That means say
something about the direction and strength of the relationship
and what it
means.