One-sample t-test: Purpose, Hypotheses, and Assumptions

7. Hypothesis Testing: One-sample t-test

One-sample t-test: Purpose, Hypotheses, and Assumptions

One of the drawbacks of using a $Z$ -test to test hypotheses about a population mean $\mu$ is that it requires more information than is usually available.

Specifically, we need to know the value of the population standard deviation $\sigma$ in order to calculate the $Z$ -statistic:
$Z = \cfrac{\bar{X} - \mu_0}{\sigma_{\bar{X}}} = \cfrac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}$

This information is very rarely available in 'real-life', however, which drastically limits the practical applicability of the $Z$ -test.

When we want to perform a hypothesis test for a population mean $\mu$ , but the value of $\sigma$ is unknown, we will have to use a one-sample Student's $t$ -test instead.
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Student's t-test

A Student's $\boldsymbol{t}$ -test is any statistical test for which the distribution of the test statistic follows a Student's $t$ -distribution under the null hypothesis.

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One-sample t-test: Purpose and Hypotheses

A one-sample $\boldsymbol{t}$ -test is used to test hypotheses about an unknown population mean $\mu$ . It is used instead of the one-sample $Z$ -test at times when the population standard deviation $\sigma$ is unknown.

Here, one-sample indicates that a single sample is analyzed to draw inferences about a single population.

The hypotheses of a one-sample $t$ -test for a population mean $\mu$ are identical to the hypotheses of a one-sample $Z$ -test for $\mu$ :

Two-tailed	Left-tailed	Right-tailed
$H_0: \mu = \mu_0$ $H_a: \mu \neq \mu_0$	$H_0: \mu \geq \mu_0$ $H_a: \mu \lt\mu_0$	$H_0: \mu \leq \mu_0$ $H_a: \mu \gt\mu_0$

Assumptions of the One-Sample t-test

The following assumptions are required to hold in order for a one-sample t-test to produce valid results:

Random sampling is used to draw the samples.
Independence of observations, meaning the occurrence of one observation does not influence the chances of another observation occurring.
The sampling distribution of the sample mean is approximately normal. The one-sample $t$ -test is quite robust to violations of normality, meaning that the assumption of normality can be slightly violated and the test will still produce valid results.