Finding the p-value for a one-tailed test
The LAX weapon-detection hypothesis test involved a random sample of size n from a population with probability π of success (detection of weapon). The data collected were x successes, and we tested the hypotheses...
H0 : π = π0
HA : π < π0
where π0 was the constant of interest (e.g. 0.80 in this example). The following steps were followed to obtain the p-value for the test.
The diagram below illustrates these steps
The telepathy example was similar, but the alternative hypothesis involved high values of π and the p-value was found by counting upper tail probabilities.
Finding the p-value for a two-tailed test
The appropriate tail probability to use depends on the alternative hypothesis. If the alternative hypothesis allows either high or low values of x, the test is called a two-tailed test,
H0 : π = π0
HA : π ≠ π0
The p-value is then double the smaller tail probability since values of x in both tails of the binomial distribution would provide evidence for HA.
Somali blood groups
In a study of sab bondsmen, a population sub-group in Northern Somalia, blood tests were conducted on a sample of 54, in order to investigate whether they differed genetically from the main population of 'free-born noble Somali'.
It is known that a proportion 0.574 of free-born noble Somali have blood group O. (Actually 574 had blood group O in a sample of 1000, but this sample size was large enough to provide a reasonably accurate estimate.) Is there any evidence that the sample proportion with blood group O in the sab bondsmen, 26 out of 54, does not come from a population with π = 0.574? This can be expressed as the hypotheses
H0 : π = 0.574
HA : π ≠ 0.574
We would expect (0.574 × 54) = 31 of the sab bondsmen to have blood group O. A sample count that is either much greater than 31 or much less than 31 would suggest a genetic difference between the sab bondsmen and the free-born noble Somali. Use the slider below to obtain the p-value.
The probability of getting as few as 26 is 0.1084. Since this is a 2-tailed test, we must also take account of the probability of getting a count that is as unusually high, so the p-value is twice this, 0.2169. Getting 26 sab bondsmen with blood group O is therefore not unlikely, so we conclude that there is no evidence from these data of a genetic difference between sab bondsmen and the free-born Somali.