Lecture 7 - Correlation, Simple Linear Regression, R-Squared

Question 1

Correlation, 𝑟_𝑥𝑦

Correlation DOES NOT imply causation

Answer 1

Rescaled version of covariance (same sign), which lies in the interval [-1, 1]

If |𝑟_𝑥𝑦| = 1, we say it has a perfectly linear relationship.

Question 2

Question

The correlation between 𝑥 and 𝑦 is 1. This implies:

A) We have a positive relationship
B) We have a perfectly linear relationship
C) We have a strong relationship
D) All of the above
E) None of the above

Answer 2

B) We have a perfectly linear relationship

Question 3

Question

The correlation between 𝑥 and 𝑦 is 0. This implies

A) We have no relationship
B) We have no linear relationship
C) x or y is zero
D) Both x and y are zero
E) None of the above

Answer 3

B) We have no linear relationship

Question 4

Simple Linear Regression

Regression Line

Answer 4

• Referred to as “line of best fit” or least-squares regression line
• Found by minimizing the sum of the squared vertical distances between each data point and the line (called residuals)

Uses of this line are to:
1. Identify associations
2. Make predictions

Question 5

Simple Linear Regression

Regression Line Equation

Answer 5

Question 6

Notes

Regression Line

Answer 6

Question 7

R-Squared, 𝑟²

coefficient of determination

Answer 7

The squared correlation, 𝑟² = 𝑟_𝑥𝑦², is a statistic called the coefficient of determination

• Describes the fraction (percentage) of variability in the data which is explained by the regression model.
• If 𝑟² is large (~80%), then it’s a good model and regression line will give solid predictions

Question 8

𝑟_𝑥𝑦

Answer 8

Question 9

𝑟_𝑖

Answer 9

Question 10

𝑟²

Answer 10

Question 11

Question

Answer 11

A) Determine whether the association is linear or non-linear

A simple linear regression assumes a linear relationship between the variables. It cannot determine whether the relationship is non-linear; for that, you would need to use other methods such as plotting the data or fitting non-linear models