Chapter 12.3, Problem 33E

During oil drilling operations, components of the drilling assembly may suffer from sulfide stress cracking. The article “Composition Optimization of High-Strength Steels for Sulfide Cracking Resistance Improvement” (Corrosion Science, 2009: 2878–2884) reported on a study in which the composition of a standard grade of steel was analyzed. The following data on y = threshold stress (% SMYS) and x = yield strength (MPa) was read from a graph in the article (which also included the equation of the least squares line).

x	635	644	711	708	836	820	810	870	856	923	878	937	948
y	100	93	88	84	77	75	74	63	57	55	47	43	38

$\begin{array}{l}\sum x_i=10,576,\sum y_i=894,\sum x_i^2=8,741,264,\\ \sum y_i^2=66,224,\sum x_i{y}_i=703,192\end{array}$

1. a. What proportion of observed variation in stress can be attributed to the approximate linear relationship between the two variables?
2. b. Compute the estimated standard deviation $s_{{\hat{\beta }}_1}$.
3. c. Calculate a confidence interval using confidence level 95% for the expected change in stress associated with a 1 MPa increase in strength. Does it appear that this true average change has been precisely estimated?