Question

4. Tamoxifen is a drug often used to treat breast cancer patients. One effect of the drug is to change the levels of cortisol-binding globulin (CBG). One study attempted to see if the effect of Tamoxifen is age dependent on the patients age. The response variable Y is the change in CBG and the covariate X is age. The following summary statistics were reported. n-, 26, Σ i-. 161 3, Σ Vi-28 19. Sr,-3756.96, S,-465.34, Sr,--757.64 (a) Find the least squares estimates of the intercept and slope (b) Give the standard errors for your estimates in (a). (Using result in Question 2. (c) Construct 95% confidence intervals for the true intercept and true slope. (d) What conclusions would you draw from your results?

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	1613	281.9	3756.96	465.34	-757.64
mean	62.03846154	10.8423077	SSxx	SSyy	SSxy

sample size ,   n =   26      
here, x̅ =   62.03846154       ȳ =   10.84230769
              
SSxx =    Σ(x-x̅)² =    3756.96      
SSxy=   Σ(x-x̅)(y-ȳ) =   -757.64      
              
A)

slope ,    ß1 = SSxy/SSxx =   -0.201663047      
              
intercept,   ß0 = y̅-ß1* x̄ =   23.35317286

B)

SSE=   (Sx*Sy - S²xy)/Sx =    312.55
      
std error ,Se =    √(SSE/(n-2)) =    3.6087

estimated std error of slope =Se(ß1) =                s/√Sxx =    0.0589

estimated std error of INTERCEPT =Se(ßo) =        Se*√(1/n+x̅²/Sxx)=       3.72050

C)

confidence interval for slope              
t critical value=   t (α/2) =t(0.025,24) =    2.0638 [ excel formula =T.INV(0.025,24) ]
margin of error ,E=       t*std error =        0.121513796
lower confidence limit =        ß̂1-E =   -0.323176843  
upper confidence limit=       ß̂1+E =   -0.08014925  
confidence interval for intercept              
t critical value=   t α/2 =    2.0638
estimated std error of slope =Se(ßo) =        Se*√(1/n+x̅²/Sxx)=       3.7205
margin of error ,E=       t*std error =        7.67874
lower confidence limit =        ß̂o-E =       15.6744
upper confidence limit=       ß̂o+E =       31.03191

D) since, confidence interval of slope is negative ,so slope is significantly less than zero

and confidence interval of intercept is positive ,so intercept is significantly greater than zero