Confidence intervals

Table of Contents

Introduction Example 1 Probability distribution of the mean of normal random variables Example 2 Plot area under normal distribution Example 3 Computing confidence intervals using Student t distribution Exersizes

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details <https://www.gnu.org/licenses/>.

This code was developed by Miodrag Bolic for the book PERVASIVE CARDIOVASCULAR AND RESPIRATORY MONITORING DEVICES

% Changing the path from main_folder to a particular chapter
main_path=fileparts(which('Main_Content.mlx'));
if ~isempty(main_path)
    %addpath(append(main_path,'/Chapter2'))
    cd (append(main_path,'/Chapter1/CLT/'))
    addpath(append(main_path,'/Service'))
end
SAVE_FLAG=0; % saving the figures in a file

Introduction

Example 1 Probability distribution of the mean of normal random variables

Plot the empirical distribution of the mean of a) n=1, b) n=2 and c) n=20 uniform random variables.

close all

clear_all_but('SAVE_FLAG');

% a) Plot hystogram for n=1

n1=rand(1,2000);

figure

histogram(n1, 'Normalization','probability')

ylabel('Relative frequency of X')

xlabel('x')

title('The empirical distribution without averaging random variables')

annonation_save('a)',"Fig1.1a.jpg", SAVE_FLAG);

%annotation('textbox',...

% [0.0117142857142857 0.00952380952380952 0.0615 0.0547619047619048],...

% 'String',{'a)'},...

% 'FitBoxToText','off',...

% 'EdgeColor',[1 1 1]);

% b) Plot hystogram for n=2

n2=rand(2,2000);

m=mean(n2);

figure

histogram(m, 'Normalization','probability')

ylabel('Relative frequency of X=X_1+X_2')

xlabel('x')

title('The empirical distribution after averaging 2 variables')

annonation_save('b)',"Fig1.1b.jpg", SAVE_FLAG);

% c) Plot hystogram for n=20

n2=rand(20,2000);

m=mean(n2);

figure

histogram(m, 'Normalization','probability')

ylabel('Relative frequency of X=X_1+...+X_{20}')

xlabel('x')

title('The empirical distribution after averaging 20 variables')

annonation_save('c)',"Fig1.1c.jpg", SAVE_FLAG);

%histfit(m)

Example 2 Plot area under normal distribution

This code is from: https://www.mathworks.com/matlabcentral/answers/128788-need-help-plotting-confidence-intervals

alpha = 0.05; % significance level - change to 0.34 for 66% confidence

mu = 0; % mean

sigma = 1; % std

x = linspace(mu-5*sigma, mu+5*sigma, 500);

cutoff1 = norminv(alpha/2, mu, sigma); % Lower 95% CI is p = 0.025

cutoff2 = norminv(1-alpha/2, mu, sigma); % Upper 95% CI is p = 0.975

y = normpdf(x, mu, sigma);

xci = [linspace(mu-5*sigma, cutoff1); linspace(cutoff2, mu+5*sigma)];

yci = normpdf(xci, mu, sigma);

figure(1)

plot(x, y, '-k', 'LineWidth', 1.5)

patch(x, y, [0.9 0.9 0.9])

patch([xci(1,:) cutoff1], [yci(1,:) 0], [1 1 1])

patch([cutoff2 xci(2,:)], [0 yci(2,:)], [1 1 1])

xticklabels({})

text(-2.5,-0.02,{' \mu-2\sigma'})

text(-0.5,-0.02,{' \mu=0'})

text(1.5,-0.02,{' \mu+2\sigma'})

message = sprintf('95 %% of the area \n under the curve');

text(-1.5,0.1,message)

title('PDF of a normal distribution N(0,1)')

Example 3 Computing confidence intervals using Student t distribution

Generate a random sample of size 50 drawn from a normal population with mean 2V and standard deviation 0.5V. Compute confidence intervals at 95% confidence. This solution is from Mathworks example at https://www.mathworks.com/help/stats/tinv.html.

mu = 2;
sigma = 0.5;
n = 50;
x = normrnd(mu,sigma,n,1);

Compute the sample mean, standard error, and degrees of freedom.

xbar = mean(x);
se = std(x)/sqrt(n);
nu = n - 1;

Find the upper and lower confidence bounds for the 95% confidence interval.

conf = 0.95;
alpha = 1 - conf;
pLo = alpha/2;
pUp = 1 - alpha/2;

Compute the critical values for the confidence bounds.

crit = tinv([pLo pUp], nu);

Determine the confidence interval for the population mean.

ci = xbar + crit*se
ci = 1×2
    1.7935    2.0828

Uncertainty at 95% confidence is:

U_95=(ci(2)-xbar)/2
U_95 = 0.0723

If we replace the t-ditribution with the normal distribution at 95% confidence, we will obtain similar results:

k=2;
ci_normal = [xbar - k*se, xbar + k*se]
ci_normal = 1×2
    1.7942    2.0821

Uncertainty at 95% confidence is:

U_95_normal=(ci_normal(2)-xbar)/2
U_95_normal = 0.0720

Exersizes

Excersize 1: Repeat the analysis for Example 3 and comment on the difference in confidence intervals when a) n=5, n=15, n=200. Replace k=2 with k=1.96.

Excersize 2: Repeat the analysis for Example 3 but compute confidence intervals at 99% confidence.