%Example to understand the meaning of confidence intervals

%Generate a Gaussian population of 10000 points, with true mean=1 and
%std=2:

x=1+2*randn(10000,1);         %randn itself creates points with zero mean and variance 1.
%Multiplying by 2 and adding one shifts the mean to 1 and the std to 2.
pop_mean=mean(x);
pop_std=std(x,1);    %Use "1" to divide by n, since we are calculating the true population std
%These are not exactly 1 and 2, since x is mathematically still a sample!!!

%Now take a series of 100 random samples, each having 61 points,
%and calculate the 95% confidence intervals for the mean in each case. 

%From Holman, Table 3.7, we see that the t-value for 60 degrees of
%freedom (sample size minus 1) is found for 95% as:

t=2;
in_bracket_count=0;
for i=1:100,
    index_set = ceil(10000.*rand(61,1));
    sample=x(index_set);
    sample_mean=mean(sample);
    sample_std=std(sample); %Now we divide by n-1, since it's a sample!
    bracket_delta=t*sample_std/sqrt(61);
    low_limit=sample_mean-bracket_delta;
    hi_limit=sample_mean+bracket_delta;
    if (pop_mean>=low_limit & pop_mean<=hi_limit),
        in_bracket_count=in_bracket_count+1;
    end
end

%Now find the percentage of samples whose confidence intervals contained
%the population mean:
in_bracket_count