The code works technically. But after all the changes I made, it's not clear if it does what you want.
This line needs correction:
chi_m_L2 = m_L2 +- t_crit * sqrt(diag(cov_m_L2));
You won't make chi_m_L2 a two-element vector this way.
%% Generate the data set
m_true = [10; 100; 9.8];
sigma = 8;
t = linspace(0, 10, 10).';
y = m_true(1) + m_true(2)*t - m_true(3)*t.^2/2 + sigma*randn(10, 1);
% Construct the G matrix
G = [ones(10,1), t, -(1/2)*t.^2];
% Solve for the least squares parameters
m_L2 = (G'*G)\(G'*y);
% Calculate the model covariance matrix
cov_m_L2 = (G'*G)/sigma^2;
% Calculate the 95% confidence intervals for the model parameters
alpha = 0.05;
t_crit = tinv(1 - alpha/2, 10 - 3);
chi_m_L2 = m_L2 +- t_crit * sqrt(diag(cov_m_L2));
% Calculate the p-value for the regression
chi2_stat = sum((y - G*m_L2).^2 / sigma^2);
p_value = chi2cdf(chi2_stat, 10 - 3);
% Display the results
fprintf('Model parameters: %f %f %f\n', m_L2);
fprintf('Confidence intervals: [%f %f] [%f %f] [%f %f]\n', chi_m_L2(1, :));
fprintf('p-value: %f\n', p_value);
