Solution Found!
Solved: During oil drilling operations, components of the
Chapter 12, Problem 31E(choose chapter or problem)
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).
\(\begin{array}{r|rrrrrrrrrrrrr} x & 635 & 644 & 711 & 708 & 836 & 820 & 810 & 870 & 856 & 923 & 878 & 937 & 948 \\ \hline y & 100 & 93 & 88 & 84 & 77 & 75 & 74 & 63 & 57 & 55 & 47 & 43 & 38 \end{array}\)
\(\begin{array}{c}\sum_{ }^{ }x_i=10,576,\ \sum_{ }^{ }y_i=894,\ \sum_{ }^{ }x_i^2=8,741,264,\\ \sum_{ }^{ }y_i^2=66,224,\ \sum_{ }^{ }x_iy_i=703,192\end{array}\)
a. What proportion of observed variation in stress can be attributed to the approximate linear relationship between the two variables?
b. Compute the estimated standard deviation \(s_{\hat{\beta}_{1}}\).
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?
Questions & Answers
QUESTION:
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).
\(\begin{array}{r|rrrrrrrrrrrrr} x & 635 & 644 & 711 & 708 & 836 & 820 & 810 & 870 & 856 & 923 & 878 & 937 & 948 \\ \hline y & 100 & 93 & 88 & 84 & 77 & 75 & 74 & 63 & 57 & 55 & 47 & 43 & 38 \end{array}\)
\(\begin{array}{c}\sum_{ }^{ }x_i=10,576,\ \sum_{ }^{ }y_i=894,\ \sum_{ }^{ }x_i^2=8,741,264,\\ \sum_{ }^{ }y_i^2=66,224,\ \sum_{ }^{ }x_iy_i=703,192\end{array}\)
a. What proportion of observed variation in stress can be attributed to the approximate linear relationship between the two variables?
b. Compute the estimated standard deviation \(s_{\hat{\beta}_{1}}\).
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?
ANSWER:Step 1 of 7
Given:
\(\begin{array}{c} \sum x_{i}=10576 \\ \sum y_{i}=894 \\ \sum x_{i}^{2}=8741264 \\ \sum y_{i}^{2}=66224 \\ \sum x_{i} y_{i} \end{array}\)