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On this page, you will learn the difference between one- and two-tailedtests.

One-tailed tests Two-tailed tests Switching between one- & two-tailed tables

One-tailed tests

In the previous pages, you learned how to perform define the hypothesisfor a statistical test, then to perform a t-test to compare means. Inthe example t-testwe performed, we defined an alternate hypothesis totest whether one mean was greater than the other: > 0.

In this situation, we tested whether one mean was higher than the other.

We were not interested in whether the first mean was lower than theother, only if it was higher. So we were only interested in one side of theprobability distribution, which is shown in the image below:

In this distribution, the shaded region shows the area represented by thenull hypothesis, H0: = 0. This actually implies 0, since the onlyunshaded region in the image shows > 0. Because we were onlyinterested in one side of the distribution, or one "tail", this type of test iscalled a one-sided or a one-tailed test. When you are using tables for

probability distributions, you should make sure whether they are forone-tailed or two-tailed tests. Depending on which they are for, youneed to know how to switch to the one you need. This is all explained

below.

A one-tailed test uses an alternate hypothesis that stateseither H1: > 0ORH1: < 0, but not both. If you want to test both,

http://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/12tailed.html#1tail%231tailhttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/12tailed.html#2tail%232tailhttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/12tailed.html#switching%23switchinghttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/ttest.html#t1meanhttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/ttest.html#t1meanhttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/ttest.html#t1meanhttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/12tailed.html#1tail%231tailhttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/12tailed.html#2tail%232tailhttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/12tailed.html#switching%23switchinghttp://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/ttest.html#t1mean

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using the alternate hypothesisH1: 0, then you need to use a two-tailed test.

Two-tailed tests

We would use a two-tailed test to see if two means are different fromeach other (ie from different populations), or from the same population.

As an example, let's assume that we want to check if the pH of a streamhas changed significantly in the past year. A water sample from thestream was analyzed using a pH electrode, where six samples weretaken. It was found that the mean pH reading was 6.5 with standarddeviation sold = 0.2. A year later, six more samples were analyzed, and themean pH of these readings was 6.8 with standard deviation sold = 0.1.

Example 1

We could use a one-tailed test, to see if the stream has a higher pH thanone year ago, for which we would use the alternatehypothesis HA: prev < current. However, we may want a more rigorous test,for the hypothesis that HA: prev current. This would mean that

both HA: prev < current and HA:prev >current were satisfied, and we could besure that there is a significant difference between the means. Theprobability distribution for a 90% confidence level, two-tailed test lookslike this:

Continuing the example, we define the null hypothesis H0: prev = current,and the alternate hypothesis HA: prev current. The d.o.f. for atwo samplemean t-test is = 7.35 7, since the d.o.f. must be a whole number.The t-value for the two sample test is

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If we consult a two-tailed t-test table, for a 95% confidence limit, we find

that t7,95% = 2.36. Since tcalc > t7,95%, we reject the null hypothesis, accept thealternate hypothesis that prev current, and can say that the means aresignificantly different.

Using Tables for One- and Two-Tailed Tests

Some tables of critical t-values only give you the values for either a one-or two-tailed test, but not both. Because of this, you will need to knowhow to use one-tailed tables for two-tailed tests, and vice versa. Theconversion is actually quite simple:

Table you have Operation

One-tailed Divide Pby 2

Two-tailed Multiply Pby 2

For example, assume you have a table to a one-tail test at the 98%confidence level and want to perform a two-tailed test. For the 98%confidence level,P= 0.02. Divide Pby 2 to get 0.01, which is a 99%confidence level. So you would compare tcalc to the value from the 98%

one-tailed table, and it would be equivalent to a two-tailed test at the99% confidence level.