One-sample t-test

Definition

A one-sample t-test is a statistical method used to determine whether the mean of a single sample is significantly different from a known or hypothesized population mean. This test is particularly useful when the sample size is small and the population standard deviation is unknown, allowing for inference about the population based on sample data.

5 Must Know Facts For Your Next Test

1. The one-sample t-test is used when comparing the sample mean to a known value, often the population mean, to assess if there is a significant difference.
2. This test calculates a t-statistic by taking the difference between the sample mean and the population mean, then dividing it by the estimated standard error of the mean.
3. Degrees of freedom for a one-sample t-test are calculated as the sample size minus one (n - 1), which is important for determining the appropriate t-distribution to use.
4. The one-sample t-test assumes that the sample data are drawn from a normally distributed population, especially important when sample sizes are small (typically n < 30).
5. If the p-value obtained from the one-sample t-test is less than the chosen significance level (commonly 0.05), we reject the null hypothesis in favor of the alternative hypothesis.

Review Questions

• How does the one-sample t-test help in assessing whether a sample mean significantly differs from a population mean?
  • The one-sample t-test evaluates if there is a statistically significant difference between the sample mean and a hypothesized population mean. It does this by calculating a t-statistic that measures how far the sample mean deviates from the population mean relative to the variability within the sample. By comparing this t-statistic against critical values from the t-distribution, we can determine if any observed difference is likely due to chance or reflects a true effect.
• Discuss the importance of degrees of freedom in conducting a one-sample t-test and how they impact results.
  • Degrees of freedom in a one-sample t-test represent the number of independent values that can vary when estimating parameters. They are calculated as n - 1, where n is the sample size. The degrees of freedom affect which t-distribution we use to determine critical values and p-values. Using the correct degrees of freedom ensures that our statistical tests have accurate significance levels, which impacts our ability to correctly reject or fail to reject the null hypothesis.
• Evaluate how assumptions about normality and sample size affect the validity of results obtained from a one-sample t-test.
  • The validity of results from a one-sample t-test hinges on assumptions about normality and sample size. When working with small samples (n < 30), normality becomes critical; if data are not normally distributed, results may be unreliable. In contrast, larger samples benefit from the Central Limit Theorem, allowing for more robust conclusions even if data aren't perfectly normal. Violating these assumptions could lead to incorrect interpretations, making it essential to check for normality and consider alternative methods if assumptions are not met.

Related terms

t-distribution: A probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but with heavier tails, used in hypothesis testing and confidence intervals when dealing with small sample sizes.

null hypothesis: A statement that assumes no effect or no difference in a statistical test, serving as the starting point for hypothesis testing.

p-value: The probability of obtaining test results at least as extreme as the observed results, under the assumption that the null hypothesis is true; it helps determine statistical significance.