Lecture 6 - Linear Relationships, Scatter Plots, Covariance - Unit 2: Regression

Question 1

Suppose that you are told the class median for
the Quiz 1 grades was 80% and the IQR was
5%.
Assuming that 𝑄2 is exactly in the middle of
𝑄1 and 𝑄3, then one quarter of the class
obtained a grade between 77.5 and 82.5%.

A) TRUE
B) FALSE

Answer 1

B) False

Half the class would have grades between 77.5 - 82.5%

Question 2

Which numerical measure(s) cannot be
approximated by looking at a boxplot?

A) Mean
B) Median
C) Range
D) Interquartile range

Answer 2

A) Mean

Question 3

Linear Relationships

Relationship Between Two Variables

bivariate data where two variables are measured per unit

Answer 3

𝑥 is called the explanatory variable (independent variable)

• explains HOW the response changes from unit to unit

𝑦 is called the response variable (dependent variable)

• Measures an outcome of a study; captures the outcome of interest

Question 4

Linear Relationships - Variables

The typical amount of calories a person consumes per day and that
person’s percent of body fat

Answer 4

higher calories = higher body fat
x - explanatory variable

• number of calories

y - response variable

• % body fat

Question 5

Linear Relationships -Variables

Inches of rain in the growing season and the yield of corn in bushels
per acre

Answer 5

more rain = higher yield
x - explanatory variable

• amount of rain

y - response variable

• yield of corn

Question 6

Linear Relationships

General Approach

Answer 6

1. Make a scatter plot of all (𝑥, 𝑦) coordinates.
2. Compute numerical measures that describe the linear relationship.
   -Covariance
   -Correlation
3. Make predictions.
   -Regression line
   -R-squared

Question 7

Scatter Plots

Scatter Plot and Relationships

Chapter 3

Answer 7

Question 8

Scatter Plots

What do we notice?

Answer 8

Flipper length increases,
Body mass increases

Positive Linear Relationship

Question 9

Scatter Plots

Interpreting Scatter Plots

Answer 9

After plotting two variables on a scatter plot, we describe the overall pattern of the relationship. Specifically, we look for:

• Form: linear, curved, clusters, no pattern
• Direction: positive, negative, no direction
• Strength: how closely the points fit the “form”

And clear deviations from that pattern (i.e., outliers of the
relationship).

Question 10

Numerical Measures of Linear Relationships

Covariance, 𝑠_𝑥𝑦

Answer 10

Question 11

Covariance, 𝑠_𝑥𝑦

How does this work?

Answer 11

𝑥₁= 1 is BELOW it’s mean.
𝑦₁ = 1 is BELOW it’s mean.

Question 12

Covariance, 𝑠_𝑥𝑦

Determine the signs for the other three grids.

Answer 12

Question 13

Covariance, 𝑠_𝑥𝑦

Covariance Calculation

Answer 13

Question 14

Covariance, 𝑠_𝑥𝑦

Question

Answer 14