Two-tailed tests for differences
The hypothesis tests on the previous page were appropriate for situations with some kind of symmetry between our attitudes towards the two groups — the alternative hypothesis did not specify any sign for the difference between the population means. This type of test is a two-tailed test since test statistics in both tails of the t distribution suggest that the alternative hypothesis holds.
Examples
Question | Alternative hypothesis |
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A biologist hand-rears chicks of an endangered bird species using two different diets. Does either diet make the chicks grow faster? | Mean chick weights after 4 weeks are different for chicks reared by the two diets. |
Oestrogen levels were measured from 20 male patients who had just had heart attacks, and for a control group of 20 normal males with the same mean age. Is oestrogen level associated with heart attacks? | Mean oestrogen level is different for the heart attack and control groups. |
Two different types of nitrogen-based fertiliser are being compared, based on wheat yields from 20 plots using each fertiliser. Does either fertiliser result in higher yields? | Mean yield is different for the two fertilisers. |
One-tailed tests for differences
In other situations, we want to test whether one specific group has a higher mean than the other group. Alternatively, we may want to test whether one specific group has a lower mean than the other group. These are called one-tailed tests.
Examples
Question | Alternative hypothesis |
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A scientist wants to assess whether low concentrations of a naturally occurring compound act as a pesticide. Twenty roses bushes are sprayed with the compound and another twenty act as controls. Numbers of aphids on each bush are counted after 10 days. Does the compound reduce the mean number of aphids? | Mean aphid count after spraying is lower. |
Fifty hens are fed a diet that contains a food supplement, and another fifty receive the standard diet. Does the supplement increase the number of eggs laid over a two-week period? | Mean number of eggs is higher for the hens getting the diet. |
Test statistic and p-value
The test statistic for a 1-tailed test is identical to that for a 2-tailed test, but the p-value is obtained from only one tail of the t distribution. We illustrate below for testing the hypotheses,
H0 : μ1 = μ2
HA : μ1 > μ2
The alternative hypothesis is only supported by very small values of .
This also corresponds to small values of the test statistic t ,
so the p-value is the lower tail probability of the t distribution.
Examples
The diagram below shows how the p-value is evaluated and interpreted for a 1-tailed test.
Use the pop-up menu to examine other data sets.
Properties of p-values
We again stress that a statistical hypothesis test cannot provide a definitive answer. The randomness of sample data means that:
Simulation when the underlying means are the same (H0 is true)
The following simulation is like one on the previous page, but a 1-tailed test is used to compare the population means. Samples of size 20 are again selected from two populations, both of which are normal with mean 75 and standard deviation 8.
Take several samples and observe the variation in the resulting p-value. Again observe that the p-values are usually greater than 0.1, so we would usually conclude that there is no evidence that µ1 is higher than µ2.
However about 1/10 of the p-values are less than 0.1, 1/20 are less than 0.05 and 1/100 are less than 0.01.