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Chapter 3 – Examining Relationships
Scatterplots and Correlation - 3.1
Scatterplots: Shows a relationship
between two variables.
Response Variables:
Variable on the y-axis.
Response to a variable
Explanatory Variables: Variable on the x-axis.
Influences the response
Looking at Scatterplots:
(DFS!)
Describe For Scatter!
• Direction: Positive as x increases, y increases
Negative as x increases, y decreases
• Form: Is there a linear relationship between the
two variables?
• Strength: Do the points follow a single stream
that is tight to the line or is there
considerable spread (or variability)
around the line?
Can the NOAA predict where a hurricane will go?
•The example in the text
shows a negative
association between
central pressure and
maximum wind speed
•As the central pressure
increases, the maximum
wind speed decreases.
Calculator Tip: Diagnostics On!
Catalog – Alpha “D” – Diagnostics On - Enter
Calculator Tip: Scatterplots
L1: Explanatory Variable
L2: Response Variable
Use statplot to graph
Scientists are interested in seeing if global
temperature has been increasing. They
measured the average global temperature per
year (in Celsius). What graph should they
make?
Histogram of Global Temp.
What does the scatterplot tell us that the histogram
didn’t?
Are female oscar winners getting older?
Example #1:
Suppose you were to collect data for each pair of variables below.
Which variable is the explanatory and which is the response?
Determine the likely direction and strength of the relationship.
1. T-shirts at a store: Price of each, Number Sold
explanatory
y
100
response
D: negative
# sold
S: strong
1
$5
$50
Price of shirt
x
Example #1:
Suppose you were to collect data for each pair of variables below.
Which variable is the explanatory and which is the response?
Determine the likely direction and strength of the relationship.
2. Drivers: Reaction Time, Blood Alcohol Level
response
explanatory
y
10
D: positive
Time
S: strong
1
.01
.5
BAC
x
Example #1:
Suppose you were to collect data for each pair of variables below.
Which variable is the explanatory and which is the response?
Determine the likely direction and strength of the relationship.
3. Cars: Age of Owner, Weight of the Car
Makes no sense!!!
Example #2:
“I have never found a quantifiable predictor in 25 years of grading that was
anywhere as strong as this one. If you just graded them based on length
without ever reading them, you’d be right over 90 percent of the time.” The
table below shows the data set that Dr. Perlman used to draw his conclusions.
Carry out your own analysis of the data. Then write a few sentences in
response to each of Dr. Perlman’s conclusions.
Essay score and length for a sample of SAT essays
Words 460
422
402
365
357
278
236
201 168
156
133
Score
6
5
5
6
5
4
4
3
2
Words 114
108
100
403
401
388
320
258 236
189
128
Score
1
1
5
6
6
5
4
3
2
6
2
4
4
Words 67
697 387
355
337
325
272
150 135
73
Score
6
5
5
4
4
2
1
1
6
3
D: positive
F: Linear, one unusual point
S: strong
Example #3:
Regraph #2 with score as the dependent variable now. Do you
see any differences in the graph?
**You may want to store these lists for tomorrow…
Correlation:
“r”
Measures the
direction and
strength of the
linear relationship
(DF only)
Must be quantitative
Attributes of the Correlation
1.The correlation coefficient is a unit-less
measurement, denoted with the letter r,
and has values between -1 and 1.
2. When r = 1 all the data points form a perfect
straight line relationship with a positive slope.
3. When r = -1 all the data points form a perfect
straight line relationship with a negative
slope.
Attributes of the Correlation
4. Correlation treats x and y symmetrically:
–
The correlation of x with y is the same as the
correlation of y with x.
5. Correlation is not affected by changes in
the center or scale of either variable.
–
Correlation depends only on the z-scores, and
they are unaffected by changes in center or
scale.
Attributes of the Correlation
6. Values of r close to 0 means that the linear
relationship is weak. There is a general linear
trend, but there is a lot of variability around
that trend.
7. When r = 0 there is no relationship between the
two variables. In other words, the best fitting
line has a slope of zero.
Attributes of the Correlation
8. Outliers have a large influence on the
correlation coefficient. The correlation is NOT
resistant to outliers.
9. Correlation does not describe curved
relationships! (ONLY LINEAR)
Guidelines: How strong is the linear relationship?
0 < r < 0.3 = weak positive
0.4 < r < 0.7 = moderate positive
0.8 < r < 1 = strong positive
-0.3 < r < 0 = weak negative
-0.4 < r < -0.7 = moderate negative
-0.8 < r < -1 = strong negative
Data collected from students in Statistics classes
included their heights (in inches) and weights (in
pounds):
•If we had to put a number
on the strength, we would
not want it to depend on
the units we used.
•A scatterplot of heights
(in centimeters) and
weights (in kilograms)
doesn’t change the
shape of the pattern:
Example #4
Types of Correlation:
r=0
r=0
r = -0.3
r = 0.5
r = -0.7
r = -0.7
r = 0.5
r = -0.99
r = -0.3
r = 0.9
r = 0.9
r = -0.99
• Don’t assume the relationship is linear just
because the correlation coefficient is high.

Here the
correlation is
0.979, but the
relationship is
actually bent.
Example #5:
What is wrong with the following statements?
1.There is a strong correlation between the
gender of American workers and their income.
Gender is categorical
Example #5:
What is wrong with the following statements?
b. We found a high correlation (r = 1.09) between
students’ rating of faculty teaching and ratings
made by other faculty members.
r can’t be bigger than 1
Example #5:
What is wrong with the following statements?
c. We found a very weak correlation (r = -0.95)
which suggests little relationship between
income and hours spent at casinos.
r = -0.95 is a strong negative relationship
Example #5:
What is wrong with the following statements?
d. We found a very weak correlation (r = 0.01)
which suggests little relationship between age
and death rate.
Should be a very strong relationship!
HOW TO CALCULATE THE CORRELATION COEFFICIENT
Remember how to calculate the z-score? We used this
calculation to determine how many standard deviations
our observations was from the mean.
RECALL:
z - score = z =
x 

In this case, we were only concerned with one
variable.
Now, we are considering two variables and
each must be standardized.
Notation:
r  correlatio n
x  sample mean of x' s
xi  the i ' th observatio n of the x' s
Sx  sample standard deviation of the x' s
n  total number of observatio ns
y  sample mean of y ' s
yi  the i' th observatio n of the y ' s
Sy  sample standard deviation of the y ' s
FORMULA:
 xi  x  yi  y 
1




r

n  1  S x  S y 
Example #4:
Speed (x)
20
30
40
MPG (y)
25
35
45
Step #1: Find the following summary statistics:
n = ___
3
SPEED:
30
x  _____
10
Sx = _____
MPG
35
y  _____
10
Sy = _____
Step #2: Calculate z-scores
SPEED
Z(x1) =
20  30
Z
10
Z  1
MPG
Z(y1) =
25  35
Z
10
Z  1
PRODUCT Z(x )Z(y ) =
1
1
1
Z(x2) =
30  30
Z
10
Z0
Z(y2) =
35  35
Z
10
Z0
Z(x2)Z(y2) =
0
Z(x3) =
40  30
Z
10
Z 1
Z(y3) =
45  35
Z
10
Z 1
Z(x3)Z(y3) =
1
Step #3: Calculate the Correlation
1
1  0  1
r

3 1
1
r  ( 2)
2
r 1
Calculator Tip: Correlation
L1: Explanatory Variable
L2: Response Variable
Stat-calc-LinReg(a+bx), L1, L2
(make sure your diagnostic is on!!!)
Example #7:
Use your calculator to find the correlation to #2. Comment on
what it means.
Words 460
422
402
365
357
278
236
201 168
156
133
Score
6
5
5
6
5
4
4
3
2
Words 114
108
100
403
401
388
320
258 236
189
128
Score
1
1
5
6
6
5
4
3
2
6
2
4
4
Words 67
697 387
355
337
325
272
150 135
73
Score
6
5
5
4
4
2
1
1
r = 0.888
6
D: positive
S: strong
3
3.2 – Least-Squares Regression
Regression line: straight line that
describes the linear
relationship between an
explanatory variable and
a response variable.
LEAST SQUARES REGRESSION LINE:
• This is the best-fitting line to the data.
• The goal is to minimize the (vertical) distances
of your observations (data) from your line.
• Again, we must square the distances (like the
calculation of the variance) because some data
points will be larger than the mean (positive)
and some are smaller than the mean (negative)
and they will cancel each other out. So to
compensate, they are squared.
We can use this line to predict a response, y,
from a given explanatory variable, x.
Remember graphing??
Slope-Intercept formula for a line:
y = mx + b
slope
where m = ____________
and
y-intercept
b = ____________
In statistics, we write it yˆ  a  bx

S
ŷ  b0  b1x
y 


1.Slope: b  r
Calculate this first!
Sx 
Do you remember the SLOPE? rise  y
run
x
Facts about Least Squares Regression:
1. The distinction between explanatory and
response variables is essential (which variable
is used to predict which?).
2. It always passes through the point (x, y).
3. Correlation ‘r’ describes the direction and
strength of the straight line, but doesn’t tell
us anymore about the slope than if it is
positive or negative, or zero.
Extrapolation: Predicting outside the range
of the x values
• Here is a timeplot of the Energy Information
Administration (EIA) predictions and actual prices
of oil barrel prices. How did forecasters do?
Example #8
Wildlife researchers monitor many wildlife populations by taking
aerial photographs in order to estimate the weights of alligators.
Here is the regression line of the weights of adult alligators (in
pounds) and their lengths (in inches) based on the data collected
from captured alligators.
Predicted Weight = – 393 + 5.9(length)
a. What is the slope of the line? What does it mean?
m = 5.9
For every inch in length, it adds 5.9 pounds
in weight
Example #8
Wildlife researchers monitor many wildlife populations by taking
aerial photographs in order to estimate the weights of alligators.
Here is the regression line of the weights of adult alligators (in
pounds) and their lengths (in inches) based on the data collected
from captured alligators.
Predicted Weight = – 393 + 5.9(length)
b. What is the y-intercept of the line? What does it mean?
b = -393
If an alligator is 0 inches, then it weights 393lbs. This makes no sense!!!
Example #8
Wildlife researchers monitor many wildlife populations by taking
aerial photographs in order to estimate the weights of alligators.
Here is the regression line of the weights of adult alligators (in
pounds) and their lengths (in inches) based on the data collected
from captured alligators.
Predicted Weight = – 393 + 5.9(length)
c. Describe the relationship between weight and length of
alligators.
As the length increases, their weight
increases.
Example #8
Wildlife researchers monitor many wildlife populations by taking
aerial photographs in order to estimate the weights of alligators.
Here is the regression line of the weights of adult alligators (in
pounds) and their lengths (in inches) based on the data collected
from captured alligators.
Predicted Weight = – 393 + 5.9(length)
d. What is the predicted weight for an alligator 90 inches
long?
yˆ = a-393
 bx+ 5.9(90)
yˆ = a-393
 bx+ 531
Calculate
this
first!
yˆ = a138
 bx
lbs
Calculate this first!

S
y 


1.Slope: b  r

S

Syx 



1.Slope: b  r

Sxy 



 Calculate this first!
1.Slope: b  r
Slope formula:
Find slope first!
yˆ  a

S

y
ˆy  a  bx 1.Slope: b  r  Calculat
Sx 
Calculate this first!
s
y = y - bx
2.
Y
intercept:
a
ŷ  b0  b1x
b1  r
sx
y - bx
Our slope is always in units of y per unit of x

S
y 

 Calculate this firs
1.Slope:
b

r
Y-intercept formula:
Sx 
yˆ  a 2.
 bx
Y - intercept: a = y - bx
alculate this first!
ŷ  b0  b1x
- bx
b0  y  b1 x
Our intercept is always in units of y
Fat Versus Protein
• The regression line for
the Burger King data
fits the data well:
– The equation is
The predicted fat content for a BK Broiler chicken
sandwich (with 30 g of protein) is 6.8 + 0.97(30) = 35.9
grams of fat.
Example #9:
Is there a relationship between wine consumption (in liters) and
yearly deaths from heart disease (deaths per 100,000)? Here are
the summary statistics:
Mean wine consumption: 3,026
Mean deaths from heart disease: 191,053
SD of wine consumption: 2,510
SD of heart disease deaths: 68,396
Correlation coefficient between wine consumption and yearly deaths from
heart disease = -.0843
a. Interpret the value of the correlation coefficient in the
context of the problem.
As wine consumption increases, mean deaths
from heart disease decreases.
Example #9:
Is there a relationship between wine consumption (in liters) and
yearly deaths from heart disease (deaths per 100,000)? Here are
the summary statistics:
Mean wine consumption: 3,026
Mean deaths from heart disease: 191,053
SD of wine consumption: 2,510
SD of heart disease deaths: 68,396
Correlation coefficient between wine consumption and yearly deaths from
heart disease = -.0843
b. Calculate the least-squares regression line predicting
death rateyˆ from
wine consumption.
a  bx
68,396 


S

y
0.0843

b  r  =
Calculate
this first!


Sx 
 2510 


-2.2971
a  y  bx = 191,053–(-2.29713,026) = 198004.0991
rcept: a = y - bx
yˆ  a  bx = 198,004.0991 – 2.2971x
Example #9:
Is there a relationship between wine consumption (in liters) and
yearly deaths from heart disease (deaths per 100,000)? Here are
the summary statistics:
Mean wine consumption: 3,026
Mean deaths from heart disease: 191,053
SD of wine consumption: 2,510
SD of heart disease deaths: 68,396
Correlation coefficient between wine consumption and yearly deaths from
heart disease = -.0843
c. Use your line to predict death rate for an average
adult who consumes 4 liters of wine.
Sy 

S
Syx 


S
Syx 


yˆ =a 198,004.0991
 bx
– 2.2971x
yˆ =a 198,004.0991
 bx
– 2.2971(4)
Calculate
yˆ =a 197,994.9107
this
bxfirst!
Calculate this first!
Calculate this first!
Example #10:
Consider n pairs of numbers. Suppose
x  4, S x  3, y  2, and S y  5.
Of the following, which could be the least squares regression
line?
Slope:
(A) y = 2 + x
r can be between  Sy 
 5  1 and 1, so slope is
(B) y = -6 + 2x
b  r   r 
(C) y = -10 + 3x
 3  between
 Sx 
(D) y = 5/3 – x
5
5
(E) y = 6 – x
 b
Passes through:  x , y 
3
2 = 2 + 4 2 = 5/3 - 4
2=6- 4
26
2=2
2  -2.33
3
Calculator Tip: LSRL
L1: Explanatory Variable
L2: Response Variable
Stat-calc-LinReg(a+bx), L1, L2, vars/y-vars/Function/ Y1
Calculator Tip: Tables
2nd – window, then 2nd - graph
Example #11:
It's easy to measure the circumference of a tree's trunk, but
not so easy to measure its height. Foresters need to develop
a model for ponderosa pines that they use to predict the
tree's height (in feet) from the circumference of its trunk (in
inches):
Trunk
Diameter
Tree
Height
8
9
7
6
13
7
11
12
35
49
27
33
60
21
45
51
a. Make a scatterplot of the data and find the LSRL.
Define any variables used in this equation.
a. Make a
scatterplot of
the data and
find the
LSRL.
Define any
variables
used in this
equation.
yˆ 
 bx + 4.54133x
= a-1.31467
Where x = trunk diameter and
Calculate this first!
yˆ =predicted
a  bx tree height

S
y 

 Calculate this first!
1.Slope: b  r
Sx 
b. How strong of an association is there?
Strong, positive correlation, r = 0.88
c. They need to cut a tree down that is 10inches in diameter.
What is the predicted height of the tree?
yˆ 
 bx + 4.54133x
= a-1.31467
yˆ 
 bx + 4.54133(10)
= a-1.31467
Calculate this first!
yˆ 
 bx
= a44.10ft
Calculate this first!

S
y 


r
Syx 




r

S
x 
d.
Oops!
When they cut it down, it was actually 50ft tall. How

S

y
 were
 r much
Calculate
this first!
they off?
pt:Sax = y - bx
They were 5.9ft over what they thought it would be!
pt: a = y - bx
Residual: How close is the data to the line?
Observed y – predicted ŷ
y  yˆ
• The linear model assumes that the
relationship between the two variables is a
perfect straight line. The residuals are the
part of the data that hasn’t been modeled.
Data = Model + Residual
or (equivalently)
Residual = Data – Model
50ft
residual
• A negative residual
means the predicted
value’s too big (an
overestimate).
• A positive residual
means the predicted
value’s too small (an
underestimate).
• In the figure, the
estimated fat of the BK
Broiler chicken
sandwich is 36 g,
while the true value of
fat is 25 g, so the
residual is –11 g of
fat.
• Some residuals are positive, others are
negative, and, on average, they cancel
each other out.
• Similar to what we did with deviations, we
square the residuals and add the squares.
• The smaller the sum, the better the fit.
• The line of best fit is the line for which the
sum of the squared residuals is smallest,
the least squares line.
Residual Plot: A plot that shows the residuals
for all the data. A good line has
no pattern in the residual plot.
Calculator Tip: Residual Plot
1. Calculate the LSRL
2. Graph L1 and RESID (in list)
Example of linear residual plots

Example of curved residual plots
Not a linear model, curved
Example of fanning residual plots
Less accurate for larger x values (fanning)
Remember BK?
Example #12:
Graph the residual plot of #2 and comment on what the graph
tells you.
Slight curve, might not be a linear model, one unusual point
Reading Computer Output:
Predictor
Constant
x-variable
Coef
y-int
Slope
S=
R-Sq= r2
StDev
T
P
R-Sq(adj) =
Example #13:
The number of students taking AP Statistics at a high school during the years
of 2000-2007 is fitted with a least squares regression line. The graph of the
residuals and some computer output is as follows.
Dependent variable is: Students
Variable
Coeff
s.e.
t
p
Constant
11
6.299
1.75
0.1313
Years
13.9286
1.0506
9.25
0.0001
s = 9.758
R-sq = 93.4% R-sq(adj) = 9.24%
How many students took AP
Statistics in the year 2003?
# Students = 11 + 13.9286(Year)
How many students took AP
Statistics in the year 2003?
# Students = 11 + 13.9286(Year)
# Students = 11 + 13.9286(3)
# Students = 11 + 41.7858
# Students = 52.7858
Residual = actual – predicted
5 = actual – 52.7858
57.7858 = actual
About 58 students took AP stats in 2003
Example #14:
An important factor in the amount of gasoline a car uses is the size of the
engine. Called “displacement”, engine size measures the volume of the
cylinders in cubic inches. The regression analysis is shown.
Dependent variable is: MPG
89 total cases of which 0 are missing
R-squared = 60.9%
R-squared (adjusted) = 60%
s = 3.056 with 89 – 2 = 82 degrees of freedom
Variable
Coefficient
s.e. of Coeff
t-ratio
Constant
34.9799
1.231
28.4
Eng. Displcmt -0.066196
0.0077
-8.64
prob
0.0001
0.0001
A car you are thinking of buying is available in two different size engines, 190
cubic inches or 240 cubic inches. How much difference might this make in your
gas mileage?
240 – 190 = 50
50(-0.066196) = -3.3098
About 3 miles less per gallon
Standard Deviation of the residuals:
Used to measure the prediction
error of the line
s
 residuals
n2
2
The Residual Standard
Deviation
• The standard deviation of the residuals
measures how much the points spread
around the regression line.
• Check to make sure the residual plot has
about the same amount of scatter
throughout.
The Residual Standard
Deviation
• We don’t need to subtract the mean
because the mean of the residuals = 0
• Make a histogram or normal probability plot
of the residuals. It should look unimodal
and roughly symmetric.
• Then we can apply the 68-95-99.7 Rule to
see how well the regression model
describes the data.
• The variation in the residuals is the key to
assessing how well the model fits.
• In the BK menu items
example, total fat has
a standard deviation
of 16.4 grams. The
standard deviation
of the residuals
is 9.2 grams.
Two-variable Statistical Calculator
http://bcs.whfreeman.com/tps3e/
Exercise
3.2 & 3.3 – Correlation of Determination, Lurking Variables
2)
(r
Correlation of Determination:
How much of the y value is explained by
the x value
Assessing the Predictive Power of the Equation:
1. Correlation of Determination: r2 = the correlation
coefficient, squared
2. It is the fraction (or percent) of the variation in
the values of y that is explained by the least-squares
regression of y on x.
3. The closer r2 is to 1, the better the regression line
describes the connection between x and y – in
particular, predictions made with the equation
will be more accurate.
Example #15
The correlation between alcohol and yearly deaths from heart
disease was -0.843. What percent of the variation in the yearly
deaths from heart disease can be explained by the regression of
yearly deaths in alcohol consumption?
r = -0.843
r2 = 0.710649
71% of deaths from heart disease can be
explained by alcohol consumption.
Example #16
Is there a linear relationship between marijuana consumption and
other drug usage? For this regression, the percent of variability
in other drug usage explained by the regression of other drugs on
marijuana use as 66.5%. What is the correlation coefficient?
r2 = .665
r = 0.815475
Moderately strong, positive realtionship
Example #17
Fast Food Sandwiches: The mean serving size for fast food
sandwiches is 7.557 ounces with a standard deviation of 2.008
ounces. The mean number of calories per sandwich is 446.9 with a
standard deviation of 143. The correlation between serving size and
calories is yˆ0.849.
 a  bx

S
the
LSRL.
ˆ
y

a

bx
 y 
a. Calculate
Calculate this first!
Sx S 
b  r y  =
Calculate
this first!
0.849(143/2.008)
Sx 
= 60.46165339
a = y - bx = 446.9 – (60.467.557) = -10.00871464
ercept: a = y - bx
yˆ  a  bx = -10.0087 + 60.4617x
yˆ isathe
 bxpredicted number of calories and x is
Calculate this first!
the serving size.

 Calculate this first!
Example #17
Fast Food Sandwiches: The mean serving size for fast food
sandwiches is 7.557 ounces with a standard deviation of 2.008
ounces. The mean number of calories per sandwich is 446.9 with a
standard deviation of 143. The correlation between serving size and
calories is 0.849.
b. What percent of the variability in calories is explained
by the least squares line with serving size?
r2 = 0.8492 = 0.720801
72% of the variability in calories is explained by
serving size
Example #17
Fast Food Sandwiches: The mean serving size for fast food
sandwiches is 7.557 ounces with a standard deviation of 2.008
ounces. The mean number of calories per sandwich is 446.9 with a
standard deviation of 143. The correlation between serving size and
calories is 0.849.
c. Use this regression line to predict the average number
of calories in a 35-ounce serving. Explain if the least
squares would be appropriate to use in this situation.
yˆ  10.0087  60.4617 x
yˆ  10.0087  60.4617(35)
yˆ  2106.1508
No, extrapolation, too far away from normal values.
Example #18:
Find the correlation of determination and correlation
coefficient for #12 and explain its meaning.
Dependent variable is: Students
Variable
Coeff
s.e.
t
p
Constant
11
6.299
1.75
0.1313
Years
13.9286
1.0506
9.25
0.0001
s = 9.758
R-sq = 93.4% R-sq(adj) = 9.24%
93.4% of the variation of students that take AP Stats is
explained by the year.
r = 0.9664, Strong, positive association between the
number of AP stats students and the year.
Cautions in Making Predictions with Regression Lines:
1. If the correlation is not strong, predictions will not
be accurate.
2. Extrapolation: Do not make predictions outside of
the range for which you have data.
3. Correlation simply does not imply causation
• The correlation may be a coincidence
• Both correlation variables might be directly
influenced by some common underlying cause
Lurking Variables:
It is a variable that is not among the
explanatory or response variables, but
influences the interpretation of the
relationship.
Causation
(z = lurking variable)
X
X
Y
Y
Z
Are you
looking
hard
enough?
Example #19
There is a positive correlation between the number of
deaths by drowning and the number of ice cream cones
sold. Is this evidence that people are not heeding the
old advice to wait 2 hours after eating before swimming
and are paying the price for it?
No! Summer is the lurking variable
Example #20
Smoke Causes Coughs: A strong relationship is
found between weekly sales of firewood and weekly
sales of cough drops from September to March. Can
we conclude that smoke from the fires causes
coughs?
No! Winter is the lurking variable
Outlier:
Observation away from the other
data points
Influential Point:
Observation that drastically changes
the LSRL
• The following scatterplot shows that something
was awry in Palm Beach County, Florida, during
the 2000 presidential election…
• The red line shows the effects that one unusual
point can have on a regression:
• The extraordinarily large shoe size gives
the data point high leverage. Wherever
the IQ is, the line will follow!
Two-variable Statistical Calculator
http://bcs.whfreeman.com/tps3e/
Outlier vs. Influential