# Two-Tailed Test: Formula & Examples

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• 0:00 Significance Tests
• 1:12 The Standard Normal…
• 2:22 Null and Alternative…
• 3:31 Two-Tailed Tests of…
• 4:57 High P and Low P
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Lesson Transcript
Instructor: Tracy Payne, Ph.D.

Tracy earned her doctorate from Vanderbilt University and has taught mathematics from preschool through graduate level statistics.

When is a statistic more than just a statistic? When it is significant, of course! This lesson explains two-tailed tests, one kind of statistical significance test. See the formula to calculate the test statistic, an example, and how to interpret the result.

## Significance Tests

Tests of significance are statistical analyses designed to evaluate the probability of finding a particular value of a random variable as compared to all possible values for that same variable. Why would we want to do that? Let's go over an example:

The residents of River Creek want to evaluate changes in the pH level of the water that runs behind their homes. For the last 32 years, the normal pH value for the creek has been 5.8, but since their small town has undergone so much new construction, has this affected the pH levels in their River Creek's water supply? The high school chemistry teacher decides to find out.

In order to conduct a significance test, he needs to ensure three conditions are met:

1. The first condition is that his sample size is sufficiently large (n > 30)
2. Second that there is independence of observations, or that the result of one sample does not influence the other results.

To meet these two conditions, he takes 50 samples of water from 10 places along the river.

3. The third condition is that his data is normally distributed on a standard normal density curve. But, what does this mean?

## The Standard Normal Density Curve

The standard normal density curve represents all of the results observed for a random variable. In this example, it represents all of the pH values observed for the river.

The mean is located at the highest point because this value and values closest to the mean are the most likely observations. As values spread out incrementally from the mean, the probability of obtaining those values decreases. The p-value is the probability associated with each observation such that the probability of obtaining a value equal to the mean is large, and the probability of obtaining a value much higher or much lower than the mean is small.

The science teacher evaluates the pH level of the 50 samples and plots the values for each observation on a graph.

This graph is a visual representation of the frequencies each value is observed. The graph shows the data is normally distributed with a mean of 6.3 and a standard deviation of 0.04.

He has met the third condition of significance testing and is ready to formulate the null and alternative hypotheses, which determine if one- or two-tailed significance tests are used.

## Null and Alternative Hypotheses

Significance tests begin by assuming there is no difference between the observed value (x) and the mean (M) of all values. This is the null hypothesis (Ho). The alternative hypothesis (Ha) can take on one of three possibilities, depending on the research question:

Ha: The observed value will be greater than the mean (x > M).

Ha: The observed value will be less than the mean (x < M).

Ha: The observed value will be either greater than or less than the mean (x > M or x < M, usually written as x is not equal to M).

Each of the first two alternative hypotheses are directional tests and therefore call for a one-tailed significance test. The third does not specify a direction; it is non-directional and so calls for a two-tailed significance test.

The research question, 'Is the mean pH level for the river significantly higher or lower than in previous years?' does not specify a direction, so the science teacher will run a two-tailed significance test.

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