Residuals
How do you find the residual of a data point?
Residual = Actual - Predicted
Predicted Price = 18.617 + 103.929 * Capacity.
This is the regression equation for disk space Capacity (in megabytes) versus Price (in dollars) at a local store. What does the slope of 103.929 mean in this context?
For every 1 mb increase in capacity, the price increases by $103.929.
In slope intercept form, y=mx+b, this variable represents the y-intercept.
What is b?
Describe a correlation of -0.9 in terms of strength (weak, moderate, or strong) and direction.
Very strong, negative (slope of line is negative so line is going downward from left to right).
We summarized the relationship between x = height of a student (in inches) and y = number of steps required to walk the length of a school hallway, with the regression line y^=113.6−0.921x.
For this model, technology gives r = -0.632. Describe the relationship.
There is a moderately strong, negative, linear relationship between a students height and the number of steps required to walk the length of a school hallway.
With an actual value of 25.8 and a predicted value of 45.9, what is the residual?
-20.1
Predicted Price = 18.617 + 103.929 * Capacity.
This is the regression equation for disk space Capacity (in megabytes) versus Price at a local store. With a capacity of 2 mb, what would you expect the price to be?
$226.475
What does r = -0.75 tell us about a linear regression?
It has a fairly strong, negative correlation (the slope is negative so the line is going downward from left to right)
Sketch a data set with a correlation correlation of 0.98.
Show on paper or whiteboard
We summarized the relationship between x = height of a student (in inches) and y = number of steps required to walk the length of a school hallway, with the regression line y^=113.6−0.921x
Interpret the slope.
For every additional inch in height, the predicted number of steps to walk the length of the hallway will decrease by 0.921 steps.
What does a negative residual indicate - where is the actual value in comparison to the predicted value?
The actual is below the predicted value.
The linear model for a local store's Number of Sales people working versus Sales is as follows: Sales = 8,106 + 91.34n (where n=Number of Sales People Working). What does the slope mean in this situation?
For every increase of 1 sales person working, sales increase by $91.34.
The linear model for a local store's Number of Sales people working versus Sales is as follows:
Sales = 8,106 + 91.34n (where n=# Sales People Working)
What is the y-intercept?
$8,106
A restaurant's menu items are compared in terms of correlation. Sugar versus Calories have a correlation of 0.25. Sugar versus Protein has a correlation of -0.68. Which has a stronger correlation?
Sugar versus Protein.
We summarized the relationship between x = height of a student (in inches) and y = number of steps required to walk the length of a school hallway, with the regression line y^=113.6−0.921x
Interpret the y-intercept. Is this meaningful in context?
If a student is 0 inches tall, it will take them 113.6 steps to walk the length of the hallway. This is not meaningful because a student wouldn't be 0 inches tall.
If a residual plot of a linear regression has a pattern in the data, what does it tell you about the data set?
We should not use a linear regression, because the points are not random
The linear model for a local store's Number of Sales people working versus Sales is as follows:
Sales = 8,106 + 91.34n (where n=# of Sales People Working)
With 14 people working, what would you expect sales to be?
$9,384.76
How do you find the predicted value (y-hat) value to calculate a residual.
To find the predicted value take x for each point and substitute into your line of best fit equation.
A student says, "There was a very strong correlation of 1.22 between Sugar and Fat content." Explain the mistake made here.
Correlation must be between -1 and +1.
We summarized the relationship between x=shoe size and y=height.
For the model, technology gives r = .03. Interpret this value.
There is no relationship
The linear model for a local store's Number of Sales people working versus Sales is as follows: Sales = 8,106 + 91.34n (where n=Number of Sales People Working). If two people are working, the actual value is $8600. What is the residual?
$311.32
An analysis of Math SAT versus Verbal SAT scores gives an equation of -
Predicted Verbal SAT Score = 171.333 + 0.6943m (where m=Math SAT Score).
What would you predict someone's verbal score to be if they got a 520 on their Math section?
532.369
The linear model for a local store's Number of Sales people working versus Sales is as follows:
Sales = 8,106 + 91.34n (where n=# of Sales People Working)
What is the meaning of the y intercept, does it make sense?
No, when no one is working, there should be no sales. This is just a starting point for the data.
Two variables produce a negative correlation. How will this affect the slope?
The slope will be negative.
We summarized the relationship between x = height of a student (in inches) and y = number of steps required to walk the length of a school hallway, with the regression line y^=113.6−0.921x
For this model, technology gives r = 0.63. Interpret this value.
About 39.9% of the variability in number of steps required to walk the length of a school hallway is accounted for by the least-squares regression line with x = height (in)