One-sample t-test

Definition

A one-sample t-test is a statistical method used to determine if the mean of a single sample is significantly different from a known population mean. This test is particularly useful when the sample size is small (typically less than 30) and the population standard deviation is unknown, making it a fundamental tool in inferential statistics.

5 Must Know Facts For Your Next Test

1. The one-sample t-test calculates the t-statistic by comparing the sample mean to the known population mean, taking into account sample variability.
2. The formula for the one-sample t-test is given by: $$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$$ 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. This test assumes that the data are approximately normally distributed, especially important for smaller sample sizes.
4. The results of a one-sample t-test provide a t-value and a corresponding p-value, which help determine whether to reject or fail to reject the null hypothesis.
5. A significance level (commonly set at 0.05) is used to decide if the p-value indicates statistical significance; if p-value < significance level, reject the null hypothesis.

Review Questions

• How does the one-sample t-test differ from other types of tests like z-tests?
  • The one-sample t-test is specifically used when dealing with small sample sizes and when the population standard deviation is unknown. In contrast, a z-test can be applied when sample sizes are large (typically n ≥ 30) and when the population standard deviation is known. This distinction is crucial because using a z-test with small samples can lead to inaccurate results due to not accounting for increased variability.
• Discuss the implications of violating assumptions of normality when conducting a one-sample t-test.
  • If the assumption of normality is violated, particularly with small sample sizes, it can lead to unreliable results in a one-sample t-test. Non-normal data may cause inaccurate p-values and t-statistics, potentially leading to incorrect conclusions about whether to reject or fail to reject the null hypothesis. In such cases, alternative non-parametric tests like the Wilcoxon signed-rank test might be more appropriate.
• Evaluate how changing the significance level affects the interpretation of results in a one-sample t-test.
  • Changing the significance level affects how conservative or liberal you are in rejecting the null hypothesis in a one-sample t-test. A lower significance level (like 0.01) makes it harder to reject the null hypothesis, increasing the chances of Type II errors (failing to detect a true effect), while a higher level (like 0.10) may lead to more Type I errors (incorrectly rejecting a true null hypothesis). Thus, it's essential to choose an appropriate significance level based on the context of the study and potential consequences of errors.

Related terms

t-distribution: A probability distribution that is symmetric and bell-shaped, used in t-tests, which accounts for small sample sizes and unknown population standard deviations.

null hypothesis: A statement asserting that there is no effect or difference, which the one-sample t-test seeks to test against the alternative hypothesis.

p-value: A measure that helps determine the significance of results from statistical tests, indicating the probability of observing the data if the null hypothesis is true.