Question

Failure times of silicon wafer microchips. Refer to the National Semiconductor study of manufactured silicon wafer integrated circuit chips, Exercise 12.63 (p. 749). Recall that the failure times of the microchips (in hours) was determined at different solder temperatures (degrees Celsius). The data are repeated in the table below.

1. Fit the straight-line model $E\left(y\right)={\beta }_0+{\beta }_{1}x$to the data, where y = failure time and x = solder temperature.
2. Compute the residual for a microchip manufactured at a temperature of 149°C.
3. Plot the residuals against solder temperature (x). Do you detect a trend?
4. In Exercise 12.63c, you determined that failure time (y) and solder temperature (x) were curvilinearly related. Does the residual plot, part c, support this conclusion?

Short answer

1. From the excel output below, the straight-line model for y on x can be written as$E\left(y\right)=30855.91-191.567x$.
2. Residual value; ε = (2312.427-1,100) =1212.427
3. From the excel output, the residual plot denotes a curvilinear relationship amongst the residuals against solder temperature (x).
4. From the graph, it can be concluded that there exists a curvilinear relationship between failure time (y) and solder temperature (x) as the residual plot indicates a curvilinear relationship.

Step-by-step solution

Straight-line model

From the excel output below, the straight-line model for y on x can be written as E(y) = 30855.91-191.567x

Prediction value

The residual for a microchip manufactured at a temperature of 149°C can be computed using

$$\begin{gathered} \epsilon =(ŷ-y) \\ Forx=149,y=1,100and \\ \hat{y}=30855.91-191.567\left(149\right) \\ \hat{y}=2312.427 \\ Therefore,\epsilon =(2312.427-1,100)=1212.427 \end{gathered}$$

Residual plot

From the excel output, the residual plot denotes a curvilinear relationship amongst the residuals against solder temperature (x).

Residual plot interpretation

From the graph, it can be concluded that there exists a curvilinear relationship between failure time (y) and solder temperature (x) as the residual plot indicates a curvilinear relationship.