Chapter 11.14, Problem 89E

Refer to the three models given in Exercise 11.88. Let $R_I^2$, $R_{II}^2$, and $R_{III}^2$ denote the coefficients of determination for models I, II, and III. Are the following statements true or false?

a. $R_I^2\ge R_{II}^2$.

b. $R_I^2\ge R_{III}^2$.

c. $R_{II}^2\le R_{III}^2$

Television advertising would ideally be aimed at exactly the audience that observes the ads. A study was conducted to determine the amount of time that individuals spend watching TV during evening prime-time hours. Twenty individuals were observed for a 1-week period, and the average time spent watching TV per evening, Y, was recorded for each. Four other bits of information were also recorded for each individual: x₁ = age, x₂ = education level, x₃ = disposable income, and x₄ = IQ. Consider the three models given below:

$\begin{array}{ll}\text{Model}\text{I:}\hfill & Y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+\epsilon \hfill \\ \text{Model}\text{II:}\hfill & Y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\epsilon \hfill \\ \text{Model}\text{III:}\hfill & Y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2+\epsilon \hfill \end{array}$