Pass or fail? Binomialrelated hypothesis testing and confidence intervals using independent samples - Improving the User Experience through Practical Data Analytics - page 109

samples, it is possible that the number of people who try each task can be different,
whether planned or unplanned. If this is the case, there is a simple, but a bit more
tedious, way to ind these theoretical expected frequencies.
Consider a revised Table 4.2 , which we now label Table 4.6 . We have added a
third search engine, Microhard (M), in this illustration. (This leads us more gently
into the next section, Section 4.8 .)
Now we have 10 people using search engines N and M, but 15 people using
search engine B. To ind the table that corresponds to Table 4.3 , the table of theoreti-
cal expected frequencies, we irst ind the row totals and column totals and grand
total. The row totals are 25 and 10, respectively; the column totals are 10, 15, and
10, respectively, and the grand total is 35 (determined by adding the row totals, or
adding the column totals. If those two totals are not the same, it means you made an
arithmetic error).
To ind the theoretical expected frequency for a cell (row, column combina-
tion), you simply multiply the row total of the cell, times the column total of the
cell, then divided by the grand total. So, for the upper left cell (the “pass,” N
cell), the theoretical expected frequency is 25*10/35 = 7.14. The good news is
that you do not need to do this for the “fail” row of N. The total for N must still
come out 10, so the fail value = 10 - 7.14 = 2.86. This easy arithmetic can be
performed for any cell. The resulting theoretical expected frequencies are shown
in Table 4.7 .
You can see that the row totals and column totals are the same as they are in Table 4.6 .
This will always be the case, no matter how different the numbers of people are per
search engine.

Table 4.6 Table of Observed Frequencies with the Number of People in Each
Column Not the Same

TASK 1
Search Eng:

N B M

Pass 3 14 8

Fail 7 1 2

Table 4.7 Table of Theoretical Expected Frequencies with the Number of People
in Each Column Not the Same

TASK 1

Search Eng: N B M

Pass 7.14 10.72 7.14

Fail 2.86 4.28 2.86