T-Test Calculator

About T-Test Calculator

The t-test determines if a sample mean is significantly different from a hypothesized population mean.

How to use this calculator

1. Enter the sample mean, sample standard deviation, and sample size.
2. Type the null hypothesis mean, \(\mu_0\).
3. Choose the type of test, left-tailed, right-tailed, or two-tailed, if the calculator asks for it.
4. Click calculate to get the t statistic, and use it to judge how far your sample mean is from \(\mu_0\).

The formula explained

The formula computes the t statistic, which measures how many estimated standard errors the sample mean is away from the hypothesized mean. A larger absolute value of \(t\) means stronger evidence against the null hypothesis.

• \(\bar{x}\) = sample mean
• \(\mu_0\) = hypothesized population mean under the null hypothesis
• \(s\) = sample standard deviation
• \(n\) = sample size
• \(t\) = test statistic

Step by step method

1. Find the difference between the sample mean and the hypothesized mean, \(\bar{x} - \mu_0\).
2. Compute the standard error, \(s / \sqrt{n}\).
3. Divide the difference by the standard error to get the t statistic.

Worked example

Problem. A teacher claims the average test score is 72. A sample of 16 students has a mean score of 75 and a standard deviation of 8. Find the one-sample t statistic.

1. Substitute into the formula, \(t = \frac{75 - 72}{8 / \sqrt{16}}\).
2. Compute the standard error, \(8 / \sqrt{16} = 8 / 4 = 2\), then the numerator, \(75 - 72 = 3\).
3. Divide, \(t = 3 / 2 = 1.5\).

Tips and common mistakes

• Use the sample standard deviation, not the population standard deviation, for a one-sample t-test.
• If your sample size is small, the t distribution is especially important because it accounts for extra uncertainty.

Frequently asked questions