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STRENGTH:
CORRELATION = r
 xi  x
1
Correlation  r 


n 1
 sx
  yi  y 

  
  sy 
Is there a relationship between student quiz
grades and their test grade?
Quiz Avg. 75 86 92 95 80
Test Avg. 79 86 100 95 90
Describe the association’s Form, Direction, and Strength
95 80 x  85.6
Test Avg. 79 86 100 95 90 y  90
Quiz Avg. 75 86
92
 xi  x
1
r


n 1
 sx
r
sx  8.26
s y  8.09
  yi  y 

  
  sy 
1  75  85.6   79  90   86  85.6   86  90   92  85.6   100  90 



 

 


5  1  8.26   8.09   8.26   8.09   8.26   8.09 
 95  85.6   95  90   80  85.6   90  90  




 
 8.26   8.09   8.26   8.09  
r = ¼ [1.7449 + -0.0239 + 0.9577 + 0.7033 + 0]
r = ¼ (3.382) = 0.8455
WARM-UP
Just put these data points into List 1,2,3,& 4
Describe the Form, Direction, and Strength of each.
A.
X
Y
0
B.
X
Y
0
0
50
1
5
1
48
2
11
2
30
3
20
3
40
4
32
4
20
5
58
5
35
6
120
6
10
7
200
7
12
r = 0.88
r = -0.87
Chapter 7 (continued)
FACTS ABOUT CORRELATION
1. Positive r refers to positively associated variables while
negative r refers to negatively associated variables.
(The Pos./Neg. sign of ‘r’ matches the slope’s sign.)
2. Correlation is ALWAYS between -1 ≤ r ≤ 1. The
correlation is strong when r is close to 1 or -1 but weak when
r is close to zero.
3. r has NO UNITS.
4. Correlation is only valuable for LINEAR relationships.
5. Like the Mean and Std. Dev., Correlation is non-resistant
and is very influenced by outliers.
CHAPTER 8 - Interpreting the Least Squares Regression
Model is:
ŷ = a + bx
The
The
ŷ
b
is called y-hat and represents the Predicted y values.
is the Slope of the linear equation:
Interpreted as: For each unit increase in x the y-variable
is predicted to change b amount on average.
The a is the y-intercept of the linear equation:
Interpreted as: ”a” is the average amount of the y
variable when x = 0.
IceCream Sales = 35 + 12.50(Temperature)
R-Squared – the percent or fraction of variation in the values
of y that is explained by the least squares regression of y on x.
R-Squared – Is also called the Coefficient of Determination.
It identifies what percent of the variation in the predicted
values of y that are attributed by x. Thus (100% - R2) of the
variation in y is attributed to by other factors.
IceCream Sales = 35 + 12.50(Temperature)
Does temperature outside affect the number of ice cream
treats a store sells. R2 = 88.5%
R2: 88.5% of the variation is the predicted amount of ice
cream sales is attributed by the temperature outside.
Fat (g)
19
Calories 920
31
34
35
1300 1310 960
39
1180
26
43
1100 1260
Regression model: Calories = 785.94 + 11.14(Fat)
Slope: For every additional gram of fat the model predicts
approximately an additional 11.14 Calories in the food.
y-intercept: If the food product contained NO fat it would
still have 785.94 calories on average.
Correlation: r = 0.562 Moderate strong positive linear
relationship.
Residual plot:
plot
R2: 31.5% of the variation in the predicted amount of
calories is attributed by the amount of fat.
Homework: Page 189: 7, 8, 9, 10, [13, 14 omit c]
Fast food is often considered unhealthy because of the
amount of fat and calories in it. Does the amount of Fat
content contribute to the number of calories a food product
contains?
Fat (g)
19
Calories 920
31
34
35
1300 1310 960
39
1180
26
43
1100 1260
1. Construct and Describe the Scatterplot for this data.
2. Find the Regression model and interpret the slope, yintercept, correlation, residual plot, and R2.