Example 12.3 Suppose the relationship between applied stress x an time-to-failur

Question

Example 12.3 Suppose the relationship between applied stress x an time-to-failure y is described by the simple linear regression model with true regression line y 65 1.2x and o 8. Then for any fixed d value x" of stress, time-to-failure has a normal distribution with mean value 65 1.2x and standard deviation 8. In the population consisting of all x, y) points, the magnitude of a typical deviation from the true regression line is about 8. For x 20, Y has mean value Aur20 65 1.2(20) SO 41 P(Y 50 when x 20 Z 1- p( (rounded to two decimal places) 0,1292 Because 25 35 Y 50 when x 25 P Z (rounded to two decimal places) 0.030 These probabilities are illustrated as the shaded areas in the figure below. PY> 50 when 20) 292 PY 50 when 25) 030 50 True regression line 65 -1.2x 20 25

Explanation / Answer

For x = 20, Y has mean value, µY.20 = 65 – 1.2(20) = 41

P(Y > 50 when x = 20) = P(Z > (50 - 41)/8) = 1 - (1.13)

µY.25 = 35, P(Y > 50 when x = 25) = P(Z > (50 - 35)/8) = 1 - (1.88)

V(Y1 – Y2 ) = 2 + 2 = 82 + 82 = 128

Note: All the answers that go in the boxes are in bold.

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