What if the data is highly skewed and $n < 30$?

The t-test is invalid. You must use a non-parametric alternative, such as the Wilcoxon signed-rank test.

When should I use a two-tailed vs. one-tailed test?

Use a two-tailed test to check for any difference ($\mu eq \mu_0$). Use a one-tailed test only if testing for a specific direction ($ $) specified prior to data collection.

Can I use a t-test if the population standard deviation $\sigma$ is known?

No. If $\sigma$ is known exactly, you must use a one-sample z-test.

How to run a one-sample t-test

Q: What if the data is highly skewed and $n < 30$?

The t-test is invalid. You must use a non-parametric alternative, such as the Wilcoxon signed-rank test.

Q: When should I use a two-tailed vs. one-tailed test?

Use a two-tailed test to check for any difference ($\mu eq \mu_0$). Use a one-tailed test only if testing for a specific direction ($ $) specified prior to data collection.

Q: Can I use a t-test if the population standard deviation $\sigma$ is known?

No. If $\sigma$ is known exactly, you must use a one-sample z-test.

A one-sample t-test compares the mean of a single sample to a hypothesized population mean. It applies when the population standard deviation is unknown and the sample is drawn from a normally distributed population or the sample size is sufficiently large ( $n \ge 30$ ).

The setup

Define the null hypothesis ( $H_0: \mu = \mu_0$ ) and the alternative hypothesis ( $H_a: \mu eq \mu_0$ , $< \mu_0$ , or $> \mu_0$ ). Choose a significance level, typically $\alpha = 0.05$ . Identify the sample size ( $n$ ), sample mean ( $\bar{x}$ ), and sample standard deviation ( $s$ ).

The steps

Compute the test statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ . 2. Determine the degrees of freedom: $df = n - 1$ . 3. Find the p-value or critical t-value using a t-distribution table. 4. Compare the test statistic to the critical value. 5. Reject $H_0$ if the p-value $< \alpha$ or if the test statistic falls in the rejection region.

Checking the result

Verify that the sample data does not contain severe outliers using a boxplot. Ensure the sample size justifies the assumption of normality via the Central Limit Theorem if the underlying distribution is not perfectly normal.

Common errors

Using a z-test instead of a t-test when the population standard deviation is unknown is a standard mistake. Another error is incorrectly substituting the sample variance ( $s^2$ ) for the sample standard deviation ( $s$ ) in the denominator.

Worked example

A machine is supposed to fill bottles with 500 mL of water. A sample of 16 bottles has a mean volume of 495 mL and a standard deviation of 8 mL. Test if the machine is underfilling at $\alpha = 0.05$ .

$H_0: \mu = 500$ . $H_a: \mu < 500$ . $n = 16$ , $\bar{x} = 495$ , $s = 8$ , $\mu_0 = 500$ . Compute standard error: $SE = \frac{s}{\sqrt{n}} = \frac{8}{\sqrt{16}} = 2$ . Compute t-statistic: $t = \frac{\bar{x} - \mu_0}{SE} = \frac{495 - 500}{2} = -2.5$ . Degrees of freedom: $df = 16 - 1 = 15$ . The critical value for a one-tailed test at $\alpha = 0.05$ with $df = 15$ is $-1.753$ . Since $-2.5 < -1.753$ , we reject $H_0$ . The machine is underfilling.

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References: OpenStax Introductory Statistics, Chapter 9 · Khan Academy Unit: Significance tests (hypothesis testing)