Definition
A two-tailed test in statistics is a method used in hypothesis testing to determine if a sample mean is significantly different from a known population mean. The test is concerned with deviations in both directions, meaning it checks if the sample mean is either significantly higher or lower than the population mean.
Explanation
In a two-tailed test, the null hypothesis (\(H_0\)) usually states that there is no effect or no difference and the alternative hypothesis (\(H_A\)) indicates that there is an effect or difference, without specifying the direction. This test helps in identifying whether the effect could plausibly occur in either direction at given levels of significance (e.g., 0.05).
Types of Hypotheses
Null Hypothesis (\(H_0\))
The null hypothesis proposes that there is no significant difference between the sample mean and the population mean, or that any observed difference is due to random variation.
Alternative Hypothesis (\(H_A\))
The alternative hypothesis suggests that there is a significant difference between the sample mean and the population mean.
Special Considerations
Significance Level
The significance level (\(\alpha\)) for a two-tailed test is typically set at 0.05, split between the two tails of the distribution, thus allocating 0.025 to each tail.
Critical Values
Critical values for a two-tailed test correspond to the points that mark the boundaries of the rejection region. For a standard normal distribution, the critical values can be found using Z-scores or T-scores depending on the sample size and variance known.
Example
Suppose a researcher wants to test whether a new drug has an effect on blood pressure that is different from the existing drug. They use a sample mean from a group of patients:
- Define Hypotheses:
- Set Significance Level:
- Find Critical Values:
For a Z-test, the critical values at \(\alpha = 0.05\) are \(\pm 1.96\).
- Compute Test Statistic:
Compare the computed test statistic with the critical values to determine if the null hypothesis can be rejected.
Historical Context
The concept of two-tailed tests was popularized by Sir Ronald A. Fisher in the early 20th century. It has since become a cornerstone of statistical hypothesis testing.
Applicability
Two-tailed tests are widely used in various fields such as psychology, medicine, and economics where the direction of the effect is not known or when testing for differences in either direction is crucial.
Comparisons
Two-Tailed vs. One-Tailed Test
- Two-Tailed Test: Tests for deviations in both directions.
- One-Tailed Test: Tests for deviations in one direction only.
Related Terms
- P-value: The probability of obtaining an observed effect, assuming the null hypothesis is true.
- Type I Error: Incorrectly rejecting the null hypothesis.
- Type II Error: Failing to reject the null hypothesis when it is false.
FAQs
What is the main advantage of a two-tailed test?
When should I use a two-tailed test?
References
- Fisher, R. A. (1935). The Design of Experiments.
- Lehmann, E. L. (1959). Testing Statistical Hypotheses.
Summary
A two-tailed test is a fundamental statistical method for hypothesis testing that allows researchers to determine if there are significant deviations in either direction from a specified population parameter. Understanding its application, assumptions, and implications helps in conducting robust and valid statistical analyses.
