Student's t-Test for Small Sample Means

The Student's t-Test is a statistical hypothesis test used when the sample size is small ( $n < 30$ ) and the population standard deviation $\sigma$ is unknown.

t-Statistic Formula:

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \quad (df = n - 1)

Where:

$\bar{x}$ is the sample mean.
$\mu_0$ is the hypothesized population mean.
$s$ is the sample standard deviation.

Example of Student's t-Test

Sample Dataset

[12, 15, 11, 14, 13]

Test $\mu_0 = 10$ at $\alpha = 0.05$

Step 1 & 2: Formulate Hypotheses & Compute Stats

\bar{x} = 13.0, \quad s \approx 1.5811, \quad df = 4

Step 3 & 4: Compute t & Compare with Critical Value

t = \frac{13.0 - 10}{0.7071} = 4.2426

|t_{\text{calc}}| = 4.24 > 2.776

Conclusion

\boxed{\text{Reject } H_0 \text{ at } \alpha = 0.05}

Therefore, because $|t_{\text{calc}}| = 4.2426 > 2.776$ , we reject the null hypothesis and conclude that the mean is significantly greater than 10.

Interactive Student's t-Test Calculator

One-Sample Student's t-Test Solver

PreviousZ-Test (Significance)

Unit 5•26 of 28

NextChi-Square Test

Student's t-Test for Small Sample Means

The Student's t-Test is a statistical hypothesis test used when the sample size is small ( $n < 30$ ) and the population standard deviation $\sigma$ is unknown.

t-Statistic Formula:

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \quad (df = n - 1)

Where:

$\bar{x}$ is the sample mean.
$\mu_0$ is the hypothesized population mean.
$s$ is the sample standard deviation.

Example of Student's t-Test

Sample Dataset

[12, 15, 11, 14, 13]

Test $\mu_0 = 10$ at $\alpha = 0.05$

Step 1 & 2: Formulate Hypotheses & Compute Stats

\bar{x} = 13.0, \quad s \approx 1.5811, \quad df = 4

Step 3 & 4: Compute t & Compare with Critical Value

t = \frac{13.0 - 10}{0.7071} = 4.2426

|t_{\text{calc}}| = 4.24 > 2.776

Conclusion

\boxed{\text{Reject } H_0 \text{ at } \alpha = 0.05}

Therefore, because $|t_{\text{calc}}| = 4.2426 > 2.776$ , we reject the null hypothesis and conclude that the mean is significantly greater than 10.

Interactive Student's t-Test Calculator

One-Sample Student's t-Test Solver

PreviousZ-Test (Significance)

Unit 5•26 of 28

NextChi-Square Test

Student's t-Test for Small Sample Means

Example of Student's t-Test

Step 1 & 2: Formulate Hypotheses & Compute Stats

Step 3 & 4: Compute t & Compare with Critical Value

Conclusion

Interactive Student's t-Test Calculator

One-Sample Student's t-Test Solver

NETZ Engine

Student's t-Test for Small Sample Means

Example of Student's t-Test

Step 1 & 2: Formulate Hypotheses & Compute Stats

Step 3 & 4: Compute t & Compare with Critical Value

Conclusion

Interactive Student's t-Test Calculator

One-Sample Student's t-Test Solver