Applying the general properties of p-values to different tests
P-values for all hypothesis tests have the properties that were described earlier in this section. You should now be able to interpret any p-value if you know the null and alternative hypotheses that it tests. (A statistical computer program is generally used to perform hypothesis tests, so knowing the details of how the p-value is obtained is of little importance.)
Example
The following data have been collected. Are they sampled from a normally distributed population?
41.9 90.6 29.9 10.2 33.7 26.9 88.5 6.5 16.6 19.2 12.6 32.0 3.6 8.1 |
68.1 57.9 -3.0 42.2 14.5 25.7 28.1 78.4 126.2 42.0 66.6 20.6 54.6 31.7 |
2.3 45.5 55.5 37.2 51.6 97.1 80.3 41.1 7.3 31.0 30.2 1.7 27.0 38.0 |
144.9 27.8 121.9 26.0 -11.5 15.5 16.9 27.3 23.9 61.1 68.2 10.0 37.8 77.1 |
24.3 63.2 -0.6 1.0 12.1 134.5 53.8 60.4 9.0 -6.4 31.0 -2.8 114.6 19.8 |
11.5 39.6 59.0 20.7 37.3 23.1 32.7 13.0 70.6 87.3 -3.2 -20.8 119.1 -0.1 |
104.4 -4.6 72.5 7.7 31.4 36.9 47.2 74.7 29.1 70.5 77.7 81.0 191.8 1.6 |
-0.8 59.4 -2.2 -12.5 81.6 44.0 63.6 114.3 33.6 83.0 70.8 50.1 55.8 28.3 |
-7.9 51.3 37.7 48.3 88.9 59.4 126.9 35.0 51.0 91.1 -2.7 79.2 0.1 12.9 |
16.2 23.0 22.4 64.4 10.2 7.6 27.7 8.0 23.5 25.3 22.5 |
The diagram below shows a histogram of the data and the best-fitting normal distribution. Could the skewness in the data have occurred by chance from a normal population?
The Shapiro-Wilkes W test can be used to test whether data come from a normal distribution:
H0 : population distribution is normal
HA : population distribution is not normal
Computer software reports the p-value for this test as "under 0.01". We conclude that the probability of obtaining such a non-normal looking sample from a normal distribution is less than 0.01, so there is strong evidence that the data do not come from a normal population.