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!


Regression equation for Usability Data

 

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.