#GOF for midterm scores with 73 data points

 x=c(10,20,30,40,50,60,70,80,90)   # 10 intervals' cut point
 z=(x-35.93)/27.33   
z
Fz=pnorm(z)
Fz
a<-c(Fz[-1],1)
a
p<-c(Fz[1],a-Fz)
E=73*p     
 E
 O=c(11, 16, 6,11,5,7,7,4,3,3)
x2=sum((O-E)^2/E)
x2
 df=length(E)-1-2    # why -2? 2 unknown par.'s.........
df
alpha=0.05
qchisq(1-alpha,df)  # chisquare table using R
x2 > qchisq(.95,df)   # Is "Ho: score is normally distributed" rejected at level 0.05?
x2 > qchisq(.9,df)    # Is "Ho: score is normally distributed" rejected at level 0.1?

#E[9]<5, E[10]<5    may combine two cells  
e=c(E[1:8],E[9]+E[10])
e
 o=c(O[1:8],O[9]+O[10])
 o
xx2=sum((o-e)^2/e)
dfx=length(e)-1-2          # why -2?    you know........
 xx2 > qchisq(.95,dfx)   # Is "Ho: score is normally distributed" rejected at level 0.05?
 xx2 > qchisq(.9,dfx)   # Is "Ho: score is normally distributed" rejected at level 0.05?

# ####      R's  OUTPUT  with above commands   #### #
> #GOF for midterm scores with 73 data points
> 
>  x=c(10,20,30,40,50,60,70,80,90)   # 10 intervals' cut point
>  z=(x-35.93)/27.33   
> z
[1] -0.9487742 -0.5828760 -0.2169777  0.1489206  0.5148189  0.8807172
[7]  1.2466154  1.6125137  1.9784120
> Fz=pnorm(z)
> Fz
[1] 0.1713677 0.2799884 0.4141129 0.5591919 0.6966602 0.8107645 0.8937307
[8] 0.9465749 0.9760589
> a<-c(Fz[-1],1)
> a
[1] 0.2799884 0.4141129 0.5591919 0.6966602 0.8107645 0.8937307 0.9465749
[8] 0.9760589 1.0000000
> p<-c(Fz[1],a-Fz)
> E=73*p     
>  E
 [1] 12.509844  7.929310  9.791086 10.590766 10.035189  8.329616  6.056532
 [8]  3.857624  2.152330  1.747702
>  O=c(11, 16, 6,11,5,7,7,4,3,3)
> x2=sum((O-E)^2/E)
> x2
[1] 14.00259
>  df=length(E)-1-2        # why -2? 2 unknown par.'s.........
> df
[1] 7
> alpha=0.05
> qchisq(1-alpha,df)  # chisquare table using R
[1] 14.06714
> x2 > qchisq(.95,df)   # Is "Ho: score is normally distributed" rejected at level 0.05?
[1] FALSE
> x2 > qchisq(.9,df)    # Is "Ho: score is normally distributed" rejected at level 0.1?
[1] TRUE
> 
> #E[9]<5, E[10]<5    may combine two cells
> e=c(E[1:8],E[9]+E[10])
> e
[1] 12.509844  7.929310  9.791086 10.590766 10.035189  8.329616  6.056532
[8]  3.857624  3.900032
>  o=c(O[1:8],O[9]+O[10])
>  o
[1] 11 16  6 11  5  7  7  4  6
> xx2=sum((o-e)^2/e)        # why -2?    you know........
> dfx=length(e)-1-2
>  xx2 > qchisq(.95,dfx)   # Is "Ho: score is normally distributed" rejected at level 0.05?
[1] TRUE
>  xx2 > qchisq(.9,dfx)   # Is "Ho: score is normally distributed" rejected at level 0.05?
[1] TRUE
>