Definition
A test statistic is a value computed from sample data that is used to test a hypothesis about a population parameter. The test statistic is central to the process of hypothesis testing, specifically in determining whether to reject the null hypothesis. The outcome of a hypothesis test hinges on whether the test statistic falls within a certain critical range. A common criterion is if the p-value associated with the test statistic is smaller than the predetermined significance level (α), or if the test statistic itself meets a threshold value.
Examples
- Z-Statistic: Used in the context of normal distribution, particularly for large sample sizes, to determine how far away a sample mean is from the population mean.
- t-Statistic: Utilized when the sample size is small and the population standard deviation is unknown. It follows a t-distribution.
- Chi-Square Statistic: Applied in tests of independence or goodness-of-fit, involving categorical data.
- F-Statistic: Used in ANOVA tests to compare variances within different groups.
Frequently Asked Questions
What is the purpose of a test statistic?
The purpose of a test statistic is to provide a basis for making decisions about the null hypothesis. By comparing the test statistic to critical values or by examining the p-value, researchers can decide whether to reject the null hypothesis.
How is a test statistic calculated?
A test statistic is calculated based on the sample data and the specific hypothesis being tested. Different types of test statistics have different formulas, often involving the sample mean, sample variance, and sample size.
What is the significance level in hypothesis testing?
The significance level, often denoted by α, is the probability of rejecting the null hypothesis when it is actually true. Common significance levels are 0.05, 0.01, and 0.10.
What is a p-value?
A p-value represents the probability of observing the test statistic, or something more extreme, assuming that the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis.
What does it mean if a test statistic is “statistically significant”?
If a test statistic is statistically significant, it means that the evidence is strong enough to reject the null hypothesis at the chosen significance level.
Can test statistics be negative?
Yes, some test statistics, such as the t-statistic, can be negative, which indicates the direction of the deviation from the hypothesized parameter.
Related Terms
- Null Hypothesis (H0): A statement that there is no effect or no difference, and it serves as the default assumption in hypothesis testing.
- P Value: A measure that indicates the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.
- Hypothesis: An assertion or conjecture about a parameter that can be tested with statistical analysis.
- Statistic: A single measure, such as a mean or standard deviation, calculated from a sample.
- Statistically Significant: A result that is unlikely to have occurred by chance, judged by a predefined threshold (usually the significance level).
Online References
Suggested Books
- “Introduction to the Practice of Statistics” by David S. Moore, George P. McCabe, and Bruce A. Craig
- “Applied Statistics and Probability for Engineers” by Douglas C. Montgomery and George C. Runger
- “The Elements of Statistical Learning: Data Mining, Inference, and Prediction” by Trevor Hastie, Robert Tibshirani, and Jerome Friedman
Fundamentals of Test Statistic: Statistics Basics Quiz
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