$1.$ Related to problem $21$ in chapter $12$ and problem $27$ in chapter $14.$ Description of the data set is provided in page $161($problem $12$ in chapter $8).$ Dataset $$EX$0812.$txt$$ is posted.

$($Note: If you use infile and input statement to read in data, put $$$$$ after the variable sex to indicate that the values are characters. E$.$g$.$

input id x$1$ x$2$ y clrzne sex $ ;$)$

a$.$ Fit the regression model of Y $($change in refraction error$)$ on X$1,$ sex and X$1*$sex$.$ Write down the fitted regression model.

b$.$ Based on the fitted model in $($a$),$ write down the regression models of Y on X$1$ for males and females separately. What is the difference of intercepts in these two models? What is the difference of slopes in these two models? Which of the parameters in model $($a$)$ do they correspond to$,$ respectively?

c$.$ Test whether the straight lines for males and females coincide $($i$.$e$.,$ are the same$)$ with alpha$=0\mathrm{.}1.$ State your null hypothesis in terms of the coefficients. Provide you SAS code and output for conducting the analysis. Figure out your F value and p value based on output, and then draw a conclusion.

d$.$ Test whether the straight lines for males and females are parallel with alpha$=0\mathrm{.}1.$ State your null hypothesis in terms of the coefficients. Provide you SAS code and output for conducting the analysis. Figure out your F value and p value based on output, and then draw a conclusion.

e$.$ Based on your answers to questions $($c$)$ and $($d$),$ comment on differences and similarities in the change in refraction$-$baseline refractive error relationship for males and females.