AP Statistics · Unit 2 · Exploring Two-Variable Data

A ceramics engineer wants to understand how the peak firing temperature inside a kiln affects the percent linear shrinkage of ceramic tiles (the percent decrease in tile length after firing, compared to before firing). Twelve tile blanks of identical size, cut from the same clay batch, were randomly assigned to be fired at different peak temperatures ranging from $950\,^{\circ}\text{C}$ to $1170\,^{\circ}\text{C}$. After firing, the percent linear shrinkage of each tile was measured. The scatterplot shows the data, along with the least-squares regression line $\hat{y} = -13.24 + 0.01927x$, where $x$ = peak firing temperature (in $^{\circ}\text{C}$) and $\hat{y}$ = predicted percent linear shrinkage (in $\%$).

(a) Describe the relationship between peak firing temperature and percent linear shrinkage shown in the scatterplot.
(b) The tile fired at a peak temperature of $1090\,^{\circ}\text{C}$ had a percent linear shrinkage of $6.6\%$. What is the residual for this tile? Explain what the sign of the residual indicates about this tile's shrinkage.
(c) Interpret the slope of the least-squares regression line in the context of this study.
(d) A separate tile blank was fired at a peak temperature of $1400\,^{\circ}\text{C}$, well above the temperatures used to collect these data. A kiln technician wants to use the regression equation $\hat{y} = -13.24 + 0.01927x$ to predict this tile's percent linear shrinkage. Is this an appropriate use of the regression line? Justify your answer.