Scatterplots

3 Concepts
Simple Linear Regression

5 Concepts
Analysis of Variance (ANOVA)

6 Concepts
Summary of Fit

3 Concepts
Parameter Estimates

1 Concept
Multiple LInear Regression (Leonard)

2 Concepts
Confidence Intervals (Part 2)

3 Concepts
Prediction Intervals

2 Concepts
**R ^{2}** shows us how much

**Scenario**: Crammer Nation University wants to develop a regression equation to predict the "Number of Recruits" a given fraternity will receive this rush season given the "Parties Thrown" by the fraternity the previous year. They take a sample of 6 fraternities on campus, resulting in the following scatterplot with line of best fit.

The purpose of each Sum of Squares is...

~~SSE is the~~*unexplained*variation.**SSR is the***explained*variation.**SST is the***total*variation. (unexplained + explained)

**SSR** = (-3.8)^{2} + (-3.0)^{2} + (-1.4)^{2} + (+1.1)^{2} + (+2.7)^{2} + (+4.4)^{2}**SSR** = (14.44) + (9.00) + (1.96) + (1.21) + (7.29) + (19.36)**SSR** = 53.26

**SST** = (-5)^{2} + (-1)^{2} + (-4)^{2} + (+3)^{2} + (+5)^{2} + (+2)^{2}**SST** = (25) + (1) + (16) + (9) + (25) + (4)**SST** = 80

**R ^{2}** =

**Answer:** 66.575% of the variation in "Number of Recruits" can be explained by the "Parties Thrown" by the fraternity the previous year.

We want to strive for R^{2} to be as close to 1.00 as possible!

The purpose of each Sum of Squares is...

**SSE is the***unexplained*variation.~~SSR is the~~*explained*variation.**SST is the***total*variation. (unexplained + explained)

**SSE** + **SSR** = **SST**

**SSE** = (-1.2)^{2} + (+2.0)^{2} + (-2.6)^{2} + (+1.9)^{2} + (+2.3)^{2} + (-2.4)^{2}**SSE** = (1.44) + (4.00) + (6.76) + (3.61) + (5.29) + (5.76)**SSE** = 26.86

**R ^{2}** = 1 - (26.86 / 80)

*Slight variation (≈) due to simplicity of this scenario for ease of learning.*

For this data point...

**SSR**is the variation*explained*by our regression.**SSE**is the variation left*unexplained*(a.k.a. due to "error").**SST**is the*total*variation. (unexplained + explained)

When we do this for all of the data points...

**R ^{2}** =

...and get the following result above...

**R ^{2}** = 0.66575

For all data points, our regression (**SSR**) is explaining 66.575% of the total variation (** SST**) in the response variable.

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