One-sample t-test

Definition

A one-sample t-test is a statistical procedure used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean. This test is particularly useful when dealing with small sample sizes, as it accounts for the additional variability introduced by estimating the population standard deviation from the sample itself.

5 Must Know Facts For Your Next Test

1. The one-sample t-test is applicable when the population standard deviation is unknown and the sample size is small (usually n < 30).
2. To conduct a one-sample t-test, you calculate the t-statistic using the formula: $$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$$ where $$\bar{x}$$ is the sample mean, $$\mu$$ is the population mean, $$s$$ is the sample standard deviation, and $$n$$ is the sample size.
3. The one-sample t-test assumes that the sample data are drawn from a normally distributed population; however, it is robust to violations of this assumption if the sample size is sufficiently large.
4. The result of a one-sample t-test includes a t-statistic and a corresponding p-value that indicate whether to reject or fail to reject the null hypothesis.
5. One-sample t-tests can be one-tailed or two-tailed, depending on whether you are testing for a difference in one specific direction or any difference from the population mean.

Review Questions

• How do you determine whether to reject or fail to reject the null hypothesis in a one-sample t-test?
  • To decide whether to reject or fail to reject the null hypothesis in a one-sample t-test, you compare the calculated p-value to your significance level (commonly set at 0.05). If the p-value is less than 0.05, you reject the null hypothesis, indicating that there is a significant difference between the sample mean and the hypothesized population mean. Conversely, if the p-value is greater than 0.05, you fail to reject the null hypothesis, suggesting that any observed difference could be due to random chance.
• Explain how the assumptions of normality and sample size impact the validity of a one-sample t-test.
  • The validity of a one-sample t-test relies on certain assumptions, primarily that the data are drawn from a normally distributed population. For small samples (n < 30), this assumption is critical because smaller samples are more sensitive to deviations from normality. However, if the sample size is larger (typically n ≥ 30), the Central Limit Theorem suggests that even if the underlying data are not perfectly normal, the distribution of sample means will approximate normality. This means that larger samples can tolerate some violations of normality without significantly affecting test results.
• Evaluate how changes in sample size affect the power of a one-sample t-test and its implications for hypothesis testing.
  • Increasing the sample size in a one-sample t-test enhances the power of the test, which is its ability to correctly reject a false null hypothesis. A larger sample provides more information about the population and leads to more accurate estimates of means and standard deviations. Consequently, with higher power, there's a greater likelihood of detecting true differences when they exist. However, while increasing sample size improves power, it also requires more resources and time. Therefore, researchers must balance these factors when designing their studies to ensure robust and meaningful results.

Related terms

Null Hypothesis: A statement asserting that there is no effect or difference, which is tested against the alternative hypothesis in hypothesis testing.

p-value: The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A low p-value indicates strong evidence against the null hypothesis.

Confidence Interval: A range of values derived from sample statistics that is likely to contain the population parameter, typically expressed at a certain confidence level (e.g., 95%).