One-Sample t-Test

Definition

The one-sample t-test is a statistical hypothesis test used to determine if the mean of a population is significantly different from a hypothesized or known value, when the population standard deviation is unknown. It is commonly used in the context of a single population mean using the Student's t-distribution.

5 Must Know Facts For Your Next Test

1. The one-sample t-test is used to test hypotheses about the mean of a single population when the population standard deviation is unknown.
2. The test statistic for the one-sample t-test is calculated as: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size.
3. The one-sample t-test follows a Student's t-distribution with $n-1$ degrees of freedom, where $n$ is the sample size.
4. The one-sample t-test can be used to determine if the population mean is equal to, greater than, or less than a hypothesized value.
5. Assumptions for the one-sample t-test include: the data is a random sample from a normally distributed population, the population standard deviation is unknown, and the sample size is small (typically less than 30).

Review Questions

• Explain the purpose and assumptions of the one-sample t-test.
  • The one-sample t-test is used to determine if the mean of a single population is significantly different from a hypothesized or known value, when the population standard deviation is unknown. The key assumptions are: 1) the data is a random sample from a normally distributed population, 2) the population standard deviation is unknown, and 3) the sample size is small (typically less than 30). This test follows a Student's t-distribution with $n-1$ degrees of freedom, where $n$ is the sample size.
• Describe the test statistic and decision-making process for the one-sample t-test.
  • The test statistic for the one-sample t-test is calculated as: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size. The calculated t-statistic is then compared to the critical value from the Student's t-distribution, based on the chosen significance level and degrees of freedom. If the test statistic falls in the rejection region, the null hypothesis is rejected, and the researcher can conclude that the population mean is significantly different from the hypothesized value.
• Discuss how the one-sample t-test can be used to test different types of hypotheses about the population mean.
  • The one-sample t-test can be used to test three different types of hypotheses about the population mean: 1) $H_0: \mu = \mu_0$ (the population mean is equal to a hypothesized value), 2) $H_0: \mu \geq \mu_0$ (the population mean is greater than or equal to a hypothesized value), and 3) $H_0: \mu \leq \mu_0$ (the population mean is less than or equal to a hypothesized value). The choice of hypothesis depends on the research question and the direction of the expected difference between the sample mean and the hypothesized population mean. The test statistic and decision-making process are the same, but the interpretation of the results will vary based on the specific hypothesis being tested.

Related terms

Hypothesis Testing:

The process of using statistical analysis to determine whether a hypothesis about a parameter of a population is likely to be true or false.

Student's t-Distribution: A probability distribution used when the population standard deviation is unknown, and the sample size is small. It is similar to the normal distribution but has heavier tails.

Sample Mean:

The average value of a random sample drawn from a population, used to estimate the population mean.