Chapter 13.3, Problem 28E

The viscosity (y) of an oil was measured by a cone and plate viscometer at six different cone speeds (x). It was assumed that a quadratic regression model was appropriate, and the estimated regression function resulting from the n = 6 observations was

y = −2113.0937 + 3.3684x − .01780x²

a. Estimate ${\mu }_{Y\cdot 75}$ , the expected viscosity when speed is 75 rpm.

b. What viscosity would you predict for a cone speed of 60 rpm?

c. If ${\displaystyle \sum y_i^2=8386.43,{\displaystyle \sum y_i=210.70,{\displaystyle \sum x_i{y}_i=17,002.00}}}$ , and ${\displaystyle \sum x_i^2{y}_i=1,419,780}$ , compute $SSE\left[={\displaystyle \sum y_i^2-{\hat{\beta }}_0{\displaystyle \sum \le y_i-{\beta }_1{\displaystyle \sum x_i{y}_i-{\hat{\beta }}_2{\displaystyle \sum x_i^2{y}_i}}}}\right]$ and s.

d. From part (c), SST=8386.43 − (210.70)²/6 = 987.35. Using SSE computed in part (c), what is the computed value of R²?

e. If the estimated standard deviation of ${\hat{\beta }}_2$ is $s_{{\hat{\beta }}_2}=.00226$ , test H₀: β₂ = 0 versus H_a: β₂ ≠ 0 at level .01, and interpret the result.