"Mode-mixity" refers to how much of crack propagation is attributable to the three conventional fracture modes of opening, sliding, and tearing. For plane problems, only the first two modes are present, and the mode-mixity angle is a measure of the extent to which propagation is due to sliding as opposed to opening. The article "Increasing Allowable Flight Loads by Improved Structural Modeling" (AIAA J., 2006: 376-381) gives the following data on x = mode- mixity angle (degrees) and y = fracture toughness (N/m) for sandwich panels use in aircraft construction.

a. Obtain the equation of the estimated regression line, and discuss the extent to which the simple linear regression model is a reasonable way to relate fracture toughness to mode-mixity angle.
b. Does the data suggest that the average change in fracture toughness associated with a one-degree increase in mode-mixity angle exceeds 50 N/m? Carry out an appropriate test of hypotheses.
c. For purposes of precisely estimating the slope of the population regression line, would it have been preferable to make observations at the angles 16, 16, 18, 18, 20, 20, 20, 20, 22, 22, 22, 22, 24, 24, 26, and 26 (again a sample size of 16)? Explain your reasoning.
d. Calculate an estimate of true average fracture toughness and also a prediction of fracture toughness both for an angle of 18 degrees and for an angle of 22 degrees, do so in a manner that conveys information about reliability and precision, and then interpret and compare the estimates and predictions?

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x 16.52 17.53 18.05 18.50 22.39 23.89 25.50 24.89 У 609.4 443.1 577.9 628.7 565.7 711.0 863.4 956.2 x 23.48 24.98 25.55 25.90 22.65 23.69 24.15 24.54 679.5 707.5 767.1 817.8 702.3 903.7 964.9 1047.3