Residual Plots

Probability & Statistics Honors​

​Assessing Regression Models

Residual

​Residual = Actual y - Predicted y

​https://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

Multiple Choice

$$y-hat=113.6-0.921x$$  models the relationship between x=the height of a student in inches and y=the number of steps to walk the length of a hallway. Calculate the residual for a person who is 63 inches tall and takes 59 steps.

• 3.423

• -3.423

• 5.436

• -5.436

Open Ended

y-hat=113.6-0.921x models the relationship between x=the height of a student in inches and y=the number of steps to walk the length of a hallway. Interpret the residual of 3.423 from the previous question.

​Residual Plots

​Plots the residuals on the y-axis and the explanatory variable on the x-axis

Bad Residual Plots

​https://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

​https://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

​https://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

​https://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

​https://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

Multiple Choice

Shown is a residual plot. Would a linear regression model of the data be most appropriate?

• YES
• NO

Multiple Choice

Would the linear model be appropriate?

• YES

• NO

Multiple Choice

What should a residual plot look like?

• Distinct Pattern
• Curve
• Random

​Coefficient of Determination (r²)

r² measures the percent of the variability in the response variable that is accounted for by the LSRL

​

Interpretation: "---- % of the variability in ----- is accounted for by the LSRL" ​

Roller coasters with larger heights usually go faster than shorter ones. Here is the LSRL for 9 roller coasters. The equation of the regression line for this relationship is y ̂=28.17+0.2143x where x=height in feet and y=maximum speed in mph.

r² = .887

​88.7% of the variability in roller coaster maximum speed is accounted for by our model, y-hat = 28.17 + 0.2143x where x=height in feet and y=maximum speed in mph.

Multiple Choice

Can you predict the battery life of a tablet using the price? Using the data from a sample of 15 tablets, the LSRL y-hat=4.67 + 0.0068x was calculated using x=price in dollars and y= battery life in hours. S=1.21 and r²=0.342. Which is the correct interpretation of the coefficient of determination.

• 34.2% of the variability in battery life is explained by the LSRL with x=price in dollars.

• The actual battery life in hours is typically 1.21 hours away from the battery life predicted by the LSRL

• There is a weak positive linear association between price in dollars and battery life.

• We predict the battery life of a tablet will increase 0.0068 hours for each increase of $1 in price

​Standard Deviation of Residuals (s)

s measures the size of the typical residual

​

Interpretation: "The actual ----- is typically ---- away from the amount predicted by the LSRL" ​

​

Roller coasters with larger heights usually go faster than shorter ones. Here is the LSRL for 9 roller coasters. The equation of the regression line for this relationship is y ̂=28.17+0.2143x where x=height in feet and y=maximum speed in mph.

S = 10.5

​The actual maximum speed of a roller coaster is typically 10.5 mph away from the maximum speed predicted by the model.

​

​​

Multiple Choice

Can you predict the battery life of a tablet using the price? Using the data from a sample of 15 tablets, the LSRL y-hat=4.67 + 0.0068x was calculated using x=price in dollars and y= battery life in hours. S=1.21 and r²=0.342. Which is the correct interpretation of the standard deviation of residuals.

• 34.2% of the variability in battery life is explained by the LSRL with x=price in dollars.

• The actual battery life in hours is typically 1.21 hours away from the battery life predicted by the LSRL

• We predict that a tablet that cost $0 would have a battery life of 4.67 hours.

• We predict the battery life of a tablet will increase 0.0068 hours for each increase of $1 in price

Open Ended

A LSRL is used to model the relationship between y=hurricane days and x=years since 1950. s=3.2 and r²=0.15. Interpret the standard deviation of residuals.

Open Ended

A LSRL is used to model the relationship between y=hurricane days and x=years since 1950. s=3.2 and r²=0.15. Interpret the coefficient of determination (r²)

​How to determine if a model is appropriate

• The residual plot shows no pattern

• small standard deviation of residuals

• Large coefficient of determination (r²)​