%
%  Practical session 2
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% Normal distribution's PDF, c.d.f, and inverse c.d.f.
% Testing for the mean using normal distribution (z-test)
% t-test for the mean and differences between the means
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%
x=[-5:0.1:5];
y=normpdf(x,0,1);
figure
plot(x,y)
grid on
xlabel('x')
ylabel('PDF')
title('Standard normal distribution\prime s PDF')
z=normrnd(0,1,10000,1);
whos z
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%  Name      Size                   Bytes  Class
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%  z     10000x1                    80000  double array
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%Grand total is 10000 elements using 80000 bytes
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%
mean(z)
var(z)
dedges=(max(z)-min(z))/15;
edges=[min(z):dedges:max(z)];
y1=histc(z,edges);
whos y1
%  Name      Size                   Bytes  Class
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%  y1       16x1                      128  double array
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%Grand total is 16 elements using 128 bytes
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y1=y1/(mean(y1)*16*dedges);
hold on
plot(edges+dedges/2,y1,'ro')
%
%
%
i1=find(abs(z)<1);  
i2=find(abs(z)<2);
i3=find(abs(z)<3);
p1=size(i1,1)*100/10000
p2=size(i2,1)*100/10000
p3=size(i3,1)*100/10000
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%
%
i1a=find(z<1);   
i2a=find(z<2);
i3a=find(z<3);
p1a=size(i1a,1)*100/10000
p2a=size(i2a,1)*100/10000
p3a=size(i3a,1)*100/10000
%
%
%
yc=normcdf(x,0,1);
figure
plot(x,yc)
grid on
xlabel('x')
ylabel('c.d.f.')
title('Standard normal distribution\prime s c.d.f.')
%
%
%
x1=[-3:0.5:3];
whos x1             
%  Name      Size                   Bytes  Class
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%  x1        1x13                     104  double array
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%Grand total is 13 elements using 104 bytes
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for i=1:size(x1,2)                 
yc1(i)=size(find(z<x1(1,i)),1)/10000;
end
hold on
plot(x1,yc1,'ro')
%
%
%
z1=sort(z);
pp=[1:1:10000]/10000;
figure
plot(pp,z1,'r','LineWidth',[2])
xlabel('P')
ylabel('x')
title('Inverse normal c.d.f.')
grid on
%
%
%
z2=norminv(pp,0,1);
hold on
plot(pp,z2,'b')
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%
%
mean(z)
h=ztest(z,0,1,0.05)
h=ztest(z,0,1,0.05,'left')
h=ztest(z,0,1,0.05,'right')
h=ztest(z,0,1,0.5,'right')
[h,p,ci]=ztest(z,0,1)
%
%
%
%  Student's t distribution and Student's t-test
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%
%
clear
z=normrnd(0,1,5,10000);
mz=mean(z);
stdz=std(z);
z1=sqrt(5)*mz./stdz;
%
%
%
mean(z1)
var(z1)
dedges=(max(z1)-min(z1))/20;
edges=[min(z1):dedges:max(z1)];
y1=histc(z1,edges);
whos y1
%  Name      Size                   Bytes  Class
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%  y1       21x1                      128  double array
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%
y1=y1/(mean(y1)*21*dedges);
%
figure
plot(edges+dedges/2,y1,'ro')
%
%
grid on
hold on
x=[-5:0.1:5];
y=normpdf(x,0,1);
plot(x,y)
grid on
xlabel('x')
ylabel('PDF')
plot(x,tpdf(x,4),'r')
title('Gaussian and Student\prime s t distributions')
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%
% Testing for significance using t distribution (t-test)
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%
%
mean(z(:,1))
h=ttest(z(:,1),0,0.05,'both')
[h,p,ci]=ttest(z(:,1),0)
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%
%
tinv(0.95,5)
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%
%
%
%
x = normrnd(0,1,100,1);
y = normrnd(0.5,1.2,100,1);
[h,significance,ci] = ttest2(x,y,0.05,'both','unequal')