A Multiple Regression Model with Four Independent Variables to Study $x _ { 1 }$

Essay

A multiple regression model with four independent variables to study accuracy in reading liquid crystal displays was used. The variables were
y = error percentage for subjects reading a four-digit liquid crystal display $x _ { 1 }$ = level of backlight (ranging from 0 to 122 $\mathrm { cd } / \mathrm { m } ^ { 2 }$ ) $x _ { 2 }$ = character subtense (ranging from $.025 ^ { \circ } \text { to } 1.34 ^ { \circ }$ ) $x _ { 3 }$ = viewing angle (ranging from $0 ^ { \circ } \text { to } 60 ^ { \circ }$ ) $x _ { 4 }$ =level of ambient light (ranging from 20 to 1500 lux)
The model fit to data was $Y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \beta _ { 4 } x _ { 4 } + \varepsilon$ The resulting estimated coefficient were $\hat { \beta } _ { 0 } = 1.52 , \hat { \beta } _ { 1 } = .02 , \hat { \beta } _ { 2 } = - 1.40 , \hat { \beta } _ { 3 } = .02 \text {, and } \hat { \beta } _ { 4 } = .0006 \text {. }$
a. Calculate an estimate of expected error percentage when $x _ { 1 } = 10 , x _ { 2 } = .5 , x _ { 3 } = 50 \text {, and } x _ { 4 } = 100$
b. Estimate the mean error percentage associated with a backlight level of 20, character subtense of .5, viewing angle of 10, and ambient light level of 30.
c. What is the estimated expected change in error percentage when the level of ambient light is increased by 1 unit while all other variables are fixed at the values given in part (a)? Answer for a 100-unit increase in ambient light level.
d. Explain why the answers in part ( c ) do not depend on the fixed values of $x _ { 1 } , x _ { 2 } \text { and } x _ { 3 }$
Under what conditions would there be such a dependence?
e. The estimated model was based on n=30 observations, with SST=39.2 and SSE=20.0. Calculate and interpret the coefficient of multiple determination, and then carry out the model utility test using $\alpha = .05$