Regression equation for Web Navigation (Number of Clicks and Time)
x = Number of clicks to find item in the Web site
y = Number of seconds to find item in the Web site
a = y intercept in a regression line equation
b = slope of the regression line
r = correlation of x and y
The above abbreviations of a and b are used in the text. Unfortunately, other texts use m as the slope and b as the y-intercept.
Slope of Regression Line
b = r * SDy / SDx = .80 * 58.1 / 5.8 = 8.0
Note that the slope of the regression line is equal to r times the slope of the SD line for positive correlations.
y-intercept
We know that (meanx, meany) is on the regression line. We use these values plus the slope in the equation of a line:
y-hat = a + b * x
meany = a + b * meanx
78.4 = a + 8.0 * 7.6
Solving for a:
a = 78.4 - (8.0 * 7.6) = 17.6 (approximately)
Regression line equation
Using the equation of a line:
y-hat = a + b * x
We add the calculated values of a and b
y-hat = 17.6 + 8.0 * x
Example Problem
What is the predicted navigation time if a user takes 10 clicks?
prediction = 17.6 + 8.0 * 10 = approximately 98 seconds
Note that the point (10, 98) is on the regression line!
x = Number of usage errors in completing the task
y = User rating of the usability of the design
Slope of Regression Line
b = r * SDy / SDx = -0.67 * 2.2 / 1.1 = -1.34
Note that the slope of the regression line is equal to r times the slope of the SD line. The slope of the regression line is negative because the correlation is negative.
y-intercept
We know that (meanx, meany) is on the regression line. We use these values plus the slope in the equation of a line:
y-hat = a + b * x
meany = a + b * meanx
14.2 = a + -1.34 * 1.1
Solving for a:
a = 14.2 - (-1.34 * 1.1) ~ 15.7
Note that my graph of the regression line intersects the y-axis at 16. So, my drawing is off a little, probably because the distance I drew for r * SD of y is too large.
Regression line equation
Using the equation of a line:
y-hat = a + b * x
We add the calculated values of a and b
y-hat = 15.7 + -1.34 * x
Example Problem
What is the predicted user rating if the user makes 2 errors?
prediction = 15.7 + -1.34 * 2 = approximately 13
Check that the point (2, 13) is on the regression line.