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 supermarket chain has two branches in a town. Based on the individual sales in one day, do shoppers at both branches tend to have equally large bills? | Mean bills are different in the two branches |
Material is produced by two looms in a textile mill. The number of flaws in the material from each loom is counted each day for a month. Does either loom produce material with fewer flaws? | Mean number of flaws per day are different for the two looms |
A teacher has two social studies classes and teaches each class about volcanoes in a different way. From each student's mark in a test, is there any evidence about which teaching method is better? | Mean marks are different for the two teaching methods |
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. |
A third of the employees in a software company are randomly selected for a training course. Two weeks after the course, all employees are asked to rate their satisfaction with their job on a scale of 0 to 10. Did attendance at the training course improve job satisfaction? | Mean rating after course is higher. |
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.
There is a (small) probability of getting random data that misleadingly suggest that the second group mean is higher.