Question

Short Answer Question We have a dataset with n= 10 pairs of observations (li, yi), and n n r; = 683, yi = 813, i=1 i=1 n n n _ x* = 47,405, viyi = 56,089, {y} = 66, 731. i=1 i=1 i=1 What is an approximate 95% confidence interval for the slope of the line of best fit?

Answer 1

	X	Y	XY	X²	Y²
total sum	683	813	56089	47405	66731
mean	68.3000	81.3000

sample size ,   n =   10          
here, x̅ =Σx/n =   68.300   ,   ȳ = Σy/n =   81.300  
                  
SSxx =    Σx² - (Σx)²/n =   756.10          
SSxy=   Σxy - (Σx*Σy)/n =   561.10          
SSyy =    Σy²-(Σy)²/n =   634.10          
estimated slope , ß1 = SSxy/SSxx =   561.100   /   756.1000   =   0.7421
                  
intercept,   ß0 = y̅-ß1* x̄ =   30.6147          
                  
so, regression line is   Ŷ =   30.6147   +   0.7421   *x
                  
SSE=   (Sx*Sy - S²xy)/Sx =    217.7090          
                  
std error ,Se =    √(SSE/(n-2)) =    5.2167          

....................

confidence interval for slope                  
α=   0.05              
t critical value=   t α/2 =    2.306   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    5.21667   /√   756.10   =   0.190
                  
margin of error ,E= t*std error =    2.306   *   0.190   =   0.437
estimated slope , ß^ =    0.7421              
                  
                  
lower confidence limit = estimated slope - margin of error =   0.7421   -   0.437   =   0.3046
upper confidence limit=estimated slope + margin of error =   0.7421   +   0.437   =   1.1796

CI ( 0.3046 , 1.1796)

...................

α=   0.01

confidence interval for intercept                  
t critical value=t α/2 =    3.3554   [excel function: =t.inv.2t(α/2,df) ]          
estimated std error of intercept =Se(ßo) =    Se*√(1/n+x̅²/Sxx)=       13.062      
margin of error ,E=   t*std error =        43.829      
                  
lower confidence limit = ß̂o - E =   30.6147   -   43.829   =   -13.2139
upper confidence limit= ß̂o + E =   30.6147   +   43.829   =   74.4434

CI ( -13.2139 , 74.4434)

...............

Please let me know in case of any doubt.

Thanks in advance!

Please upvote!