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
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 lecturer has two accounting classes and teaches each class about automated accounting systems 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
A plumbing firm is concerned about the time it takes some residential customers to pay for work. It wonders whether mailing reminders for each overdue account more regularly will help encourage customers to pay promptly. Half of the overdue accounts are sent monthly reminders (the current practice) and the other half are sent reminders fortnightly. Is average time to payment of the accounts reduced? Mean time to payment is lower for the fortnightly reminders
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.