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.