Question

The table shows the population P (in millions) of the United States from 1800 to 1870 where t represents the number of years since 1800. Source: U.S. Bureau of the Census 5.3 10 7.2 9.6 30 129 40 17.0 50 23.2 60 31.4 70 39.8 12. Use a graphing calculator to find an exponential growth model and a logistic growth model for the data. Then graph both models. 13. Use the models from part (a) to find the year when the population was about 92 million. Which of the models gives a year that is closer to 1910, the correct answer? Explain why you think that model is more accurate. 14. Use each model to predict the population in 2010. Which model gives a population closer to 297.7 million, the predicted population from the U.S. Bureau of the Census? 15. ANALYZING MODELS The graph of a logistic growth function e reaches its point of maximum growth In a where2, show that the x coordinate of this point is
i need 13,14 and 15 please

Answer 1

SOLUTION:

12.

Note: Since the student has not asked for question 12, I have taken the liberty of using a Matlab code instead of a graphing calculator. Hope that is okay.

Please find the code for fitting attached below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%This code fits time vs population data into exponential and logistic
%models

%time
t=0:10:70;
t=t';

P=[5.3,7.2,9.6,12.9,17.0,23.2,31.4,39.8]'; %population

f=fit(t,P,'exp1'); %exponential fitting and plotting
disp(f);
figure(1);
plot(f,t,P);
xlabel('Time in years from 1800');
ylabel('Population in millions');
title('Exponential model');

[ Qpre, D, sm, varcov] = fit_logistic(t,P); %logistic fitting
tfit=0:1:70;
Plogifit=D(2)./(1 + exp(-D(3)*(tfit-D(1))));%finding the fit data
%plotting
disp(D);
figure(2);
scatter(t,P);
hold on
plot(tfit,Plogifit);
xlabel('Time in years from 1800');
ylabel('Population in millions');
title('Logistic model');
legend('data','fitted model');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%This open source code is downloaded from https://in.mathworks.com/matlabcentral/fileexchange/41781-fit_logistic-t-q

function [ Qpre, p, sm, varcov] = fit_logistic(t,Q)
%fit a logistic function to time series Q(t).
% Inputs: t (time),Q (time series variable)
% Outputs: Qpre (logistic model fit to data) and
% p is 3 element vector containing parameters describing the logistic:
% thalf, Qinf, and alpha
% Q(t) = Qinf/(1 + exp(-alpha*(t-thalf)))
% thalf is symmetric inflection point
% Qinf is value as t --> infinity
% alpha is time decay constant
% sm is 3 element vector giving 1-sigma confidence limits of parameters
% e.g., thalf = p(1) +/- sm(1)
% simply double the values in sm to get 95% confidence limits
% varcov is the complete 3x3 variance-covariance matrix for investigating
% how model paramters co-vary with each other.
% sm is sqrt(diag(varcov))
%
% Example:
% Qinf = 10.2; alpha = 0.33; thalf = 108.5;
% t = 100:120;
% Q = Qinf./(1 + exp(-alpha*(t-thalf)));
% noise = randn(1,length(t));
% Qd = Q+noise;
% Qpre = fit_logistic(t,Qd);
% figure(1)
% clf
% hold on
% plot(t,Qd,'o') % data
% plot(t,Qpre) % best fitting logistic
%   
% Written by James Conder, Southern Illinois University, Oct. 2010
% Cleaned up for publishing May 16, 2013
% May 17, 2013: Allow for decreasing logistic.
% May 23, 2013: Fix instability when using short
% or long absolute times (relative to alpha = 1).
% May 28, 2013: added example in comments, fixed an introduced bug
% from May 23 edit.
% Feb 12, 2014: Revisited occasional flatlining problem.
% (Qpre goes to mean).
% Made initial alpha more robust. Scaled to time rather than simply
% defaulting to one (removes much of need for rescaling time).
% Added check for flatlining. If occurs, reset seeds with larger
% alpha and start over.
% Jan 25, 2016: calculate confidence limits for parameters

% equations are set up to solve for an increasing logistic.
% Check if decreasing and if so, reverse time
[~,I] = sort(t);
reverse_t = false;
if sum(diff(Q(I))) < 0 % decreasing in time
reverse_t = true;
t = -t;
end

% stretch short or long sequences in time to stabilize alpha
tstretch = [];
if max(t)-min(t) < 1.e-4 || max(t)-min(t) > 1e5;
tstretch = 1./(max(t) - min(t));
t = t*tstretch;
end

% initial guesses for parameters
thalf = 0.5*(min(t) + max(t));
Qinf = max(Q);
alpha = 1./(max(t)-min(t)); alphareset = alpha;

flipQ = false;
if isrow(Q)
flipQ = true; % expecting a column vector. flip if row.
t = t';
Q = Q';
end

itermax = 1000 ;   % number of maximum iterations
epsilon = 1;

ii = 0 ; % initialize counter
thresh = 1.e-6 ; % threshold to stop iterating
G = zeros(length(t),3) ; % dimensionalize partial derivative matrix

while epsilon > thresh
ii = ii + 1 ;
Qpre = Qinf./(1 + exp(-alpha*(t-thalf))) ; % 'predicted' data
if max(Qpre) - min(Qpre) == 0
% if Qpre flatlines, "a" likely needed to be seeded higher
% (sharper climb)
alphareset = 2*alphareset;
thalf = 0.5*(min(t) + max(t));
Qinf = max(Q);
alpha = alphareset;
Qpre = Qinf./(1 + exp(-alpha*(t-thalf))) ;
end
d = Q - Qpre ; % data vector (predicted - observed)

% linearized partial derivatives
ee = min(exp(-alpha*(t-thalf)),1.e12) ;
eee = 1./((1 + ee).^2) ;
G(:,1) = -Qinf*alpha*(ee.*eee) ; % dd/dthalf
G(:,2) = 1./(1 + ee) ; % dd/dQinf
G(:,3) = Qinf*(t-thalf).*(ee.*eee) ; % dd/dalpha
  
[U,S,V] = svd(G,0);               % Singular Value Decomposition
Sinvdiag = 1./diag(S) ;
ising = Sinvdiag(1)./Sinvdiag < 1.e-12 ;
Sinvdiag(ising) = 0;
Sinv = diag(Sinvdiag);
dm = 0.1*V*Sinv*U'*d;
                          
% get new parameters: m = m0 + dm
thalf = thalf + dm(1);
Qinf = Qinf + dm(2);
alpha = alpha + dm(3);                 
                                         
epsilon = norm(dm);
if ii > itermax
   disp('max number of iterations reached...exiting')
disp(['normalized epsilon: ' num2str(epsilon/thresh)])
epsilon = thresh ;
end
end
Qpre = Qinf./(1 + exp(-alpha*(t-thalf))) ; % 'predicted' data

if ~isempty(tstretch)
thalf = thalf/tstretch;
alpha = alpha*tstretch;
end
if reverse_t % decreasing logistic
thalf = -thalf;
alpha = -alpha;
end

if flipQ
Qpre = Qpre';
Q = Q'; % necessary for a posteriori variance
end
p = [ thalf Qinf alpha ];

%%% find confidence bounds for parameters (1-sigma)
if nargout > 2
sd = sum((Q-Qpre).^2)/(length(Q)-3); % a posteriori variance
varcov = sd*sd*inv(G'*G); % model variance-covariance matrix
sm = sqrt(diag(varcov)); % model variance
end

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

By running the code, we get the models and their graphs as given below:

1. Exponential model:
   P-5.539e(00284%
   
2. Logistic model:
   186.4511 186.4511 1+ є-0. -1 + 35.3673e_00324t 0324*(t-110.0006)
   

Exponential model 45 data fitted curve 40 35 30 c 25 20 15 10 5 0 10 20 30 40 50 60 70 Time in years from 1800

Logistic model 45 O data fitted model 40 35 30 c 25 20 15 10 5 0 10 20 30 40 50 60 70 Time in years from 1800

13, 14:

Exponential mod re k= 5.539 bt ou Fn P= 92 milion 92 0 0284 5.53 1- I8. 9428 Jn year, adding 1800 1898.9 스 18 o2 mn.

Loalskc Model we have ound that ot 1 +ae Whrre186- 4511 a 35.3673 Now I ae aYe 2 af In /a C-

Tor P- a2 milon 35, 3673 Kaz 0,032千 8 G.452 · 109.2636 t In vears, adding 860 1 Year -909.2 1gog Tu the logitic mod gives value h he txial modl, th pplatn sai vegardles ef the Potulation in the lagishc maddl, plah reaches a limit In natut, po wlaton sie is ulhimate limitd by eiource arailakilty. knce, th« logisti modtl urce arailabiliy. Rence eller and qives a more accurate rejult . 1. Erjonential model

15:

Date ae GİYEn- that arowth ataims is z In(a) Το shsu: 잇 γス But C 2 2 2. a.

NOTE: IN Q15, IT IS GIVEN THAT y = c/2 WHEN THE GROWTH IS MAXIMUM IN THE LOGISTIC MODEL. AS A BONUS, THIS IS SHOWN IN THE FOLLOWING PAGES:

PONUS PROOF: To sho that outh is mxmum n ant 2 la Foi «routh dyldi) to be moximum ere d' 01 t ae ←双 - 82 s ae

In a t ae au 2

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