Statistical Analysis for Education and Psychology Researchers

there are 9Bs 6Gs and 7 runs (runs are indicated by underlining). These data meet the
assumptions underlying the one-sample runs test, each event belongs to a dichotomy
(B/G), and the events are recorded in the sequence of their occurrence.
The research hypothesis is to determine, prior to further analysis, whether the pattern
of teachers’ interactions is a non-random process (two-tailed test). The selected alpha is
p=0.05 and the number of predominant interactions with boys is 9 (n 1 ) and the number of
interactions with girls is 6 (n 2 ). The exact sampling distribution of U (number of runs) is
used because the number in both of the categories n 1 , n 2 is <20.
Table 4 in Appendix A4 gives the critical values for U when n 1 , or n 2 ≤20. We enter
Table 4 with n 1 equals 9 and n 2 equals 6 and find that the critical values are 4 and 13. (It
makes no difference if you enter the tables with the values n 1 equals 6 and n 2 equals 9.)

Interpretation

Table 4 in Appendix A4 shows that for values of n 1 , equals 9 and n 2 equals 6, a random
sample would contain, 95 times out of 100, between 4 and 13 runs. The observed number
of runs, 7, is within this region, and we do not reject the null hypothesis of non-
randomness. The researcher is able to conclude that the sequence of teacher interactions
observed over the 15 science lessons is random.
If the researcher had decided in advance that departure form randomness would be in a
direction such that too many runs would be observed (for example, the teacher may have
been conscious of being observed and may therefore make an extra effort to interact with
girls rather than boys), then only the larger of the two values in the body of the table
should be used (13). If the observed number of runs is greater than this critical value, we
reject the null hypothesis and conclude that the alternative directional hypothesis is
tenable.
With a one-tailed test alpha would be 0.025 rather than 0.05 (for the two-tailed test).
Since the observed value is less than the upper critical value (7<13) we would not reject
the null hypothesis of non-randomness and could not conclude that there were too many
runs.

Worked Example

Large Sample Approximation

Consider the data from the previous example but assume that an additional twenty
science lessons were observed. The complete data set is now:
Lesson12345678 9 101112131415
Code BBGBBGBB B GGGGBB
Lesson 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Code GGBGGBBB G BBBBGG
Lesson3132333435
Code BBBBG

Statistical analysis for education and psychology researchers 216