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

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

1) Using model (iii.), find a 95% confidence interval for the mean Cycles to crack initiation at a PSA of 130

Answer 1

iii) The simple linear regression can be performed on excel:

Following are the steps

Step 1: Put the data into the excel spreadsheet

Step 2: Convert the normal values to log values by using (=ln) command

Step 3: Go to data --> data analysis --> regression --> select the data

Step 4: Select the data --> click ok

Based on the regression results, the equation will be:

ln (y) = 40.678 - 6.513*ln(x)

95% confidence interval for the mean cycles to crack initiation at a PSA of 130

The mean cycle value PSA of 130 is:

ln (y) = 40.678 - 6.513*ln(130)

ln (y) = 8.9575

Following is the formula for calculating the confidence interval:

Confidence interval = ln (y) +/- z*S.E.

where,

z = z score at given confidence interval

S.E. = Standard Error

The value of z at a 95% confidence interval is 1.96

The value of standard error is mentioned in the output as 0.3984

Confidence Interval = 8.9575 +/- 1.96*0.3984

Confidence Interval = 8.9575 +/- 0.7808

Confidence Interval = (8.1767,9.7383)