CI on the Mean

The Confidence Interval on the Mean uses the sample mean and sample standard deviation as estimates of the population parameters. When the number of observations n is small (≤ 30), the Student's t Distribution is used; when large, Z Scores from the Standard Normal Distribution are used. mctad.confidenceIntervalOnTheMean() 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 a random sample of n observations, x_bar is the sample mean, s is the sample standard deviation, 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.confidenceIntervalOnTheMean(x_bar, s, n, α, 'u')

Inline Comments

mctad.confidenceIntervalOnTheMean = function (x_bar, s, n, α, type) { if (typeof x_bar !== 'number' || typeof s !== 'number' || !mctad.isInteger(n) || α <= 0.0 || α >= 1.0) { return undefined; } var σ_bar = s / Math.sqrt(n);

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

if (n > 30) {

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

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

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

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

Return both bounds of a two-tailed confidence interval.

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

Otherwise, use Student's t distribution. Return the upper confidence bound of a one-tailed confidence interval.

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

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

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

Return both bounds of a two-tailed confidence interval.

return [ x_bar - mctad.t_distribution_table[n - 1][1 - α / 2] * σ_bar, x_bar + mctad.t_distribution_table[n - 1][1 - α / 2] * σ_bar ]; } } } };