Regression Analysis - Beta Values

#1
I have a question on a sample test for my upcoming prelims and I am unsure how to answer a question, scenario and questions below...

A study measured dental measurements from the center of the pituitary to the pterygomaxillary fissure (measured in mm) for 11 girls and 16 boys at ages 8, 10, 12, and 14. The subjects are 27 individual children, and there are four repeated measurements on each child. The table below lists the dental measurements for the 27 children. The data (attached) show that each measurement is only 0.01 units apart, therefore making the difference too small to analyze.

A second investigator decides to analyze 1) if the average (i.e. the variable called mean) dental measurements for ages 8, 10, 12, and 14 are associated with gender and 2) if the difference in dental measurements from age 12 to age 8 (i.e. the variable called diff) are associated with gender. The data (attached, highlighted in yellow) shows the mean but the difference as the incorrect measurement (mean + 0.015).

There is also a proof showing that the Beta for Y will be the same as the Beta for Y+c, so the Betas for the second investigator should be the same since it is still a Y+c scenario like the first investigator had in his data set. BUT the Betas are different, and I do not know why. Any ideas?

I have attached images of the questions with the datasets and the output for the second investigator, as well as the proof showing the Betas should be the same. Thanks!
 

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