Question

An experiment is performed to study the fatigue performance of a high strength alloy. The number of cycles to crack initiation is measured for twenty specimens over a range of applied pseudo-stress amplitude (PSA) levels. Use the data in the table provided to fit the following three regression models with y = Cycles and x = PSA (note the natural log transform of y for all models):

PSA (x) Cycles (y)

80, 97379

80, 340084

80, 246163

80, 239348

100, 34346

100, 23834

100, 70423

100, 51851

120, 9139

120, 9487

120, 8094

120, 17956

140, 5640

140, 3338

140, 6170

140, 5608

160, 1723

160, 3525

160, 2655

160, 1732

i. A simple linear regression model: lny=β0+β1∙x .

ii. A quadratic polynomial model: lny=γ0+γ1∙x+γ2∙x2 .

iii. A simple linear regression model with a logarithm transformation on PSA: lny=δ0+δ1∙ln⁡(x) .

1. For model (i.), what hypothesis being tested by the ANOVA F-test (in terms of the coefficients of the model)? Interpret the conclusion of this test in the context of the engineering problem.
2. For model (ii.), what hypothesis being tested by the ANOVA F-test (in terms of the coefficients of the model)? Interpret the conclusion of this test in the context of the engineering problem.
3. What is the p-value for testing the significance of the quadratic term in model (ii.) (Ho: γ2=0)? Interpret the conclusion of this test in the context of the engineering problem.
4. Briefly discuss the advantages and disadvantages of each of the three models.

Answer 1

Answer:-

Given That:-

An experiment is performed to study the fatigue performance of a high strength alloy. The number of cycles to crack initiation is measured for twenty specimens over a range of applied pseudo-stress amplitude (PSA) levels.

Since p - value of F - test = 0.000237653 < 0.05

So over all model is significant.

_______________________________________________________

Since p - value of F - test = 2.76822 x 10-6< 0.05

So over all model is significant.

Since p - value corresponding x2 = 0.000488 < 0.05

So is significantly different from zero.

_______________________________________________________

Coefficient of determination for model (i) = 0.537044908

Coefficient of determination for model (ii) = 0.778108173

Coefficient of determination for model (iii) = 0.946825259

Hence model (iii) is the best model since it explains 94.68% of total variation in y and this is maximum among three.

However three models are significant.

Thank you for your supporting. Please upvote my answer...