Two-Tailed Tests

A two-tailed test is a statistical test in which the null hypothesis is that the population mean is equal to a specific value, and the alternative hypothesis is that the population mean is not equal to that value. The test statistic is calculated by comparing the sample mean to the hypothesized population mean, and the p-value is the probability of obtaining a sample mean as extreme as or more extreme than the one observed if the null hypothesis is true.

Two-tailed tests are used when there is no prior expectation about the direction of the difference between the sample mean and the hypothesized population mean. For example, if a researcher is testing the effectiveness of a new drug, they would use a two-tailed test to determine whether the drug has a significant effect on the patient's condition, regardless of whether the effect is positive or negative.

The p-value for a two-tailed test is calculated by finding the area under the normal distribution curve that is more extreme than the test statistic. The p-value is used to determine whether the null hypothesis should be rejected. If the p-value is less than or equal to the significance level, the null hypothesis is rejected. If the p-value is greater than the significance level, the null hypothesis is not rejected.

The significance level is the probability of rejecting the null hypothesis when it is true. The most commonly used significance level is 0.05, which means that there is a 5% chance of rejecting the null hypothesis when it is true.

Two-tailed tests are more powerful than one-tailed tests, but they are also more likely to result in a Type I error. A Type I error occurs when the null hypothesis is rejected when it is true.