Chapter 13.4, Problem 48E

An experiment to investigate the effects of a new technique for degumming of silk yarn was described in the article “Some Studies in Degumming of Silk with Organic Acids” (J. Society of Dyers and Colourists, 1992: 79–86). One response variable of interest was y = weight loss (%). The experimenters made observations on weight loss for various values of three independent variables: x₁ = temperature (°C) = 90, 100, 110;x₂ = time of treatment (min) = 30, 75, 120; x₃ = tartaric acid concentration (g/L) = 0,8,16. In the regression analyses, the three values of each variable were coded as −1, 0, and 1, respectively, giving the accompanying data (the value y₈ = 19.3 was reported, but our value y₈ = 20.3 results in regression output identical to that appearing in the article).

A multiple regression model with k = 9 predictors—x₁, x₂, x₃, $x_5=x_2^2,x_6=x_3^2$, x₇ = x₁x₂, x₈ = x₁x₃, and x₉ = x₂x₃—was fit to the data, resulting in ${\hat{\beta }}_0=21.967$, ${\hat{\beta }}_1=2.8125$, ${\hat{\beta }}_2=1.2750$, ${\hat{\beta }}_3=3.4375$, ${\hat{\beta }}_4=-2.208$, ${\hat{\beta }}_5=1.867$, ${\hat{\beta }}_6=-4.208$, ${\hat{\beta }}_7=-.975$, ${\hat{\beta }}_8=-3.750$, ${\hat{\beta }}_9=-2.325$, and R² = .938

a.    Does this model specify a useful relationship? State and test the appropriate hypotheses using a significance level of .01.

b.    The estimated standard deviation of ${\hat{\mu }}_Y$ when x₁ = ⋯ = x₉ = 0 (i.e., when temperature = 100, time = 75, and concentration = 8) is 1.248. Calculate a 95% CI for expected weight loss when temperature, time, and concentration have the specified values.

c.    Calculate a 95% PI for a single weight-loss value to be observed when temperature, time, and concentration have values 100, 75, and 8, respectively.

d.    Fitting the model with only x₁, x₂, and x₃ as predictors gave R² = .456 and SSE = 203.82. Does at least one of the second-order predictors provide additional useful information? State and test the appropriate hypotheses.