CI on the Difference between Two Means

The Confidence Interval on the Difference between Two Means uses the sample means and sample standard deviations of the two sampled distributions as estimates of the population parameters. When the number of observations n_x or n_y are small (≤ 30), the Student's t Distribution is used; when large, Z Scores from the Standard Normal Distribution are used. mctad.confidenceIntervalOnTheDifferenceBetweenTwoMeans() returns an Array of lower, upper values for the (1 - α/2)interval or a Number for the lower or upper (1 - α) boundary.

Assumptions

Given samples from two distributions, x_bar and y_bar are the sample means, s_x and s_y are the sample standard deviations, n_x and n_y are the number of observations, and 0.0 < α < 1.0. By default, the confidence interval is two-tailed; this may be changed by specifying type as either 'l' for lower, or 'u' for upper. Any other value for type generates a two-tailed confidence interval.

Use

mctad.confidenceIntervalOnTheDifferenceBetweenTwoMeans(x_bar, s_x, y_bar, s_y, n_x, n_y, α, 'u')

Inline Comments

mctad.confidenceIntervalOnTheDifferenceBetweenTwoMeans = function (x_bar, s_x, y_bar, s_y, n_x, n_y, α, type) { if (typeof x_bar !== 'number' || typeof s_x !== 'number' || typeof y_bar !== 'number' || typeof s_y !== 'number' || !mctad.isInteger(n_x) || !mctad.isInteger(n_y) || α <= 0.0 || α >= 1.0) { return undefined; } var x_bar_minus_y_bar = x_bar - y_bar, σ_x_bar_minus_y_bar = Math.sqrt(Math.pow(s_x, 2) / n_x + Math.pow(s_y, 2) / n_y), v;

If the sample size is large, use Z Scores from the Standard Normal Distribution.

if (n_x > 30 && n_y > 30) {

Return the upper confidence bound of a one-tailed confidence interval.

if (typeof type !== 'undefined' && type.toLowerCase() === 'u') { return x_bar_minus_y_bar + mctad.z(1 - α) * σ_x_bar_minus_y_bar; } else {

Return the lower confidence bound of a one-tailed confidence interval.

if (typeof type !== 'undefined' && type.toLowerCase() === 'l') { return x_bar_minus_y_bar - mctad.z(1 - α) * σ_x_bar_minus_y_bar; } else {

Return both bounds of a two-tailed confidence interval.

return [ x_bar_minus_y_bar - mctad.z(1 - α / 2) * σ_x_bar_minus_y_bar, x_bar_minus_y_bar + mctad.z(1 - α / 2) * σ_x_bar_minus_y_bar ]; } } } else {

Otherwise, calculate the degrees of freedom and use Student's t distribution.

v = Math.floor( Math.pow(Math.pow(s_x, 2) / n_x + Math.pow(s_y, 2) / n_y, 2) / ((Math.pow(Math.pow(s_x, 2) / n_x, 2) / (n_x - 1)) + (Math.pow(Math.pow(s_y, 2) / n_y, 2) / (n_y - 1))) );

Return the upper confidence bound of a one-tailed confidence interval.

if (typeof type !== 'undefined' && type.toLowerCase() === 'u') { return x_bar_minus_y_bar + mctad.t_distribution_table[v][1 - α] * σ_x_bar_minus_y_bar; } else {

Return the lower confidence bound of a one-tailed confidence interval.

if (typeof type !== 'undefined' && type.toLowerCase() === 'l') { return x_bar_minus_y_bar - mctad.t_distribution_table[v][1 - α] * σ_x_bar_minus_y_bar; } else {

Return both bounds of a two-tailed confidence interval.

return [ x_bar_minus_y_bar - mctad.t_distribution_table[v][1 - α / 2] * σ_x_bar_minus_y_bar, x_bar_minus_y_bar + mctad.t_distribution_table[v][1 - α / 2] * σ_x_bar_minus_y_bar ]; } } } };