One-Sample t-Test

Hypothesis Testing

The one-sample t-test checks whether the mean of your data is significantly different from a hypothesized value.

Hypotheses:

• $H_0$ (null): $mu = mu_0$ (the mean equals the hypothesized value)
• $H_1$ (alternative): $mu eq mu_0$

import math, statistics

data = [2.1, 2.5, 2.3, 2.8, 2.4]
mu_0 = 2.0   # hypothesized mean
n = len(data)

t_stat = (statistics.fmean(data) - mu_0) / (statistics.stdev(data) / math.sqrt(n))
print(round(t_stat, 4))   # 4.0

The t-Statistic

t = rac{ar{x} - mu_0}{s / sqrt{n}}

A large $|t|$ means the sample mean is far from $mu_0$ relative to the data's variability.

The p-value

The $p$-value is the probability of observing a $t$-statistic this extreme (or more) if $H_0$ were true.

• $p < 0.05$ → statistically significant (reject $H_0$ at 5% significance level)
• $p geq 0.05$ → insufficient evidence to reject $H_0$

Special Case: Testing Against the Sample Mean

When $mu_0$ equals the sample mean exactly, $t = 0$ and $p = 1.0$.

Your Task

Implement ttest_one_sample(data, mu) that prints the $t$-statistic (rounded to 4 decimal places) and whether the result is significant ($p < 0.05$).