What a t‑Test Is and When You Use It in Statistics

What Is a t‑Test?

The t‑test evaluates:

\[ \text{Difference between sample mean and hypothesized mean} \]

or

\[ \text{Difference between two sample means} \]

scaled by the estimated standard error.

Why We Use the t‑Distribution

1. σ is unknown in real data

We rarely know the population standard deviation, so we estimate it using the sample standard deviation.

2. Estimating σ adds uncertainty

This extra uncertainty widens the tails of the distribution.

3. The t‑distribution accounts for small samples

With fewer observations, the estimate of σ is noisier, so the t‑distribution adjusts for this.

4. As n increases, t approaches normal

For large samples, the t‑test and z‑test become nearly identical.

How a t‑Test Works (Step by Step)

Step 1: State hypotheses

\[ H_0: \mu = \mu_0 \quad\text{vs}\quad H_a: \mu \neq \mu_0 \]

Step 2: Compute the test statistic

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

Step 3: Determine degrees of freedom

\[ df = n – 1 \]

Step 4: Compare to the t‑distribution

Find the p‑value or critical value.

Step 5: Make a conclusion

Reject or fail to reject \(H_0\) based on α.

Numerical Example

Sample: n = 10, mean = 52, s = 8 Test: \(H_0: \mu = 50\)

\[ t = \frac{52 – 50}{8/\sqrt{10}} = 0.79 \]

With df = 9, the p‑value is large → fail to reject \(H_0\).