Ive got myself confused over the difference when preforming a 2 tailed test opposed to the a one tailed test, does somebody have a link to a worked example/any light to throw on the subject for me? what do I do differently this time?

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- Jul 29th 2008, 10:35 AMSilenceInShadowsTwo tailed chi squared tests,
Ive got myself confused over the difference when preforming a 2 tailed test opposed to the a one tailed test, does somebody have a link to a worked example/any light to throw on the subject for me? what do I do differently this time?

- Jul 30th 2008, 04:46 AMmr fantastic
Tables of $\displaystyle \chi^2$ used for tests of significance are typically based on one tail only (the tail to the right) of the sampling distribution of $\displaystyle \chi^2$. These are NOT critical values for directional (or one-tailed) tests). Although only one tail of the sampling distribution of $\displaystyle \chi^2$ is used, the table values are those required for testing the significance of a difference regardless of direction (that is, for a two-tailed test).

For one degree of freedom:

0.05 of the area of the $\displaystyle \chi^2$ distribution falls to the right of $\displaystyle \chi^2 = 3.84$.

0.01 of the area of the $\displaystyle \chi^2$ distribution falls to the right of $\displaystyle \chi^2 = 6.64$.

These are NOT critical values for a one-tailed test.

For 1 degree of freedom the square root of $\displaystyle \chi^2$ is a normal distribution and can be used with reference to the standard normal curve in applying two-tailed tests. In effect, because the $\displaystyle \chi^2$ distribution is the square of the standard normal distribution, both tails of the normal curve are incorporated in the right tail of the $\displaystyle \chi^2$ curve. For the standard normal distribution:

Significance at the 0.05 level for a two-tailed test is z = 1.96. Note that 1.96^2 = 3.84.

Significance at the 0.01 level for a two-tailed test is z = 2.58. Note that 2.58^2 = 6.64 (within rounding error).

In tests of goodness of fit and in most tests of independence you won't usually be concerned with the direction of the difference observed. The value of $\displaystyle \chi^2$ (with one degree of freedom) required for significance at the:

0.05 level for a one-tailed test is 2.71 (this is 1.64^2 and 1.64 is the value of z required for significance at the 0.05 level for a one tailed test).

0.01 level for a one-tailed test is 5.41 (this is 2.33^2 and 2.33 is the value of z required for significance at the 0.01 level for a one tailed test).