One-Sample t-Test

Formula

\[ t = \frac{\bar x - \mu_0}{s/\sqrt{n}} \]

Plot

fn: 1/(1+exp(-5*(x-0.5)))
xmin: 0
xmax: 1.5
ymin: 0
ymax: 1.05
height: 280
title: Example power curve for one-sample t-test (illustrative)

Parameters

• \(\bar x\): sample mean
• \(\mu_0\): hypothesized mean
• \(s\): sample standard deviation
• \(n\): sample size

What it means

Tests whether the population mean differs from \(\mu_0\).

What it's used for

• Hypothesis testing of a single mean.
• Small-sample inference when variance is unknown.

Key properties

• Under \(H_0\), \(t\) follows Student's t with \(n-1\) degrees of freedom.
• Two-sided or one-sided alternatives are possible.

Common gotchas

• Assumes independent samples and roughly normal data.
• Not robust to heavy tails or strong outliers.

Example

If \(\bar x=52\), \(\mu_0=50\), \(s=6\), \(n=9\), then \(t=(52-50)/(6/3)=1\).

How to Compute (Pseudocode)

Input: data, null hypothesis H0, test statistic T
Output: test statistic and p-value decision summary

compute the observed test statistic T_obs from the data
obtain the null distribution (analytic approximation or exact table, depending on the test)
compute the p-value from the null distribution and tail convention
compare p-value to alpha (if making a decision)
return T_obs and p-value

Complexity

• Time: Depends on the specific test (summary-statistic computation is often linear in sample size; p-value computation may be constant-time with a CDF call or more expensive if resampling is used)
• Space: Depends on whether intermediate summaries or resampled/null distributions are materialized
• Assumptions: Test-specific assumptions (independence, variance structure, distributional assumptions) determine validity and exact computation details