One-Sample t-test

Definition

Core Statement

The One-Sample t-test determines whether the sample mean is statistically different from a known or hypothesized population mean ( $μ_{0}$ ). It is used when the population standard deviation is unknown.

t = \frac{\bar{x} - μ_{0}}{s / \sqrt{n}}

Purpose

Quality Control: Does the average weight of bags produced equal the target (e.g., 500g)?
Benchmarking: Does our school's average score differ from the national average?
Difference Scores: Testing if the average difference (Pre - Post) is non-zero (Paired t-test is technically a one-sample test on differences).

When to Use

Use When...

You have one group of continuous data.
You want to compare the mean to a specific value ( $μ_{0}$ ).
Population Deviation ( $σ$ ) is Unknown (use $s$ from sample).
$n < 30$ (if $n > 30$ , t converges to z, but t-test is still safe).

Assumptions

Independence: Observations are independent.
Normality: Sample data should be approximately normally distributed (or $n$ large enough for Central Limit Theorem (CLT)).

Worked Example: Potato Chips

Problem

Bags of chips claim to hold 200g ( $μ_{0} = 200$ ).
You sample 10 bags: Mean = 195g, StDev = 5g.

Calculate SE: $s / \sqrt{n} = 5 / \sqrt{10} \approx 1.58$ .
Calculate t: $t = \frac{195 - 200}{1.58} = \frac{- 5}{1.58} \approx - 3.16$
Decision:
- Degrees of freedom = $10 - 1 = 9$ .
- Critical t ( $α = 0.05$ , two-tailed) $\approx \pm 2.26$ .
- $| - 3.16 | > 2.26$ . Reject Null.

Conclusion: The bags are significantly underfilled.

Python Implementation

import scipy.stats as stats

data = [198, 195, 202, 190, 194, 199, 195, 192, 196, 189]
mu_0 = 200

t_stat, p_val = stats.ttest_1samp(data, popmean=mu_0)

print(f"T-statistic: {t_stat:.3f}")
print(f"P-value: {p_val:.4f}")

if p_val < 0.05:
    print("Reject Null: Mean is different from 200.")

Student's T-Test - The two-sample version.
Z-Test - Used if $σ$ is known.
Confidence Intervals - The interval version of this test.

One-Sample t-test

Definition

Purpose

When to Use

Assumptions

Worked Example: Potato Chips

Python Implementation

Related Concepts