Step by step
The question
Question: men: 40 yes, 60 no; women: 50 yes, 50 no: at the 5% level, is the answer linked to the group?
1 Write the hypotheses
H₀ is the "nothing new" idea.
Here: the group and the answer are independent.
Hₐ is what we want to show.
Here: the group and the answer are not independent.
Independent means no link: the answer does not depend on the group.
2 Totals
The counts in a table.
Each row's total: men, 100; women, 100.
Each column's total: yes, 90; no, 110.
And everyone: 200.
3 Expected counts
If there is no link, each cell expects its row total times its column total, over the grand total.
Men, yes: 100 times 90, over 200, is 45.
Men, no: 100 times 110, over 200, is 55.
Women, yes: 100 times 90, over 200, is 45.
Women, no: 100 times 110, over 200, is 55.
4 Each part
Chi-square adds up, for each cell, the observed minus the expected, squared, over the expected.
A table: each cell, what we saw, O, and what we expected, E.
40 minus 45 is −5.
Squared, 25.
Over 45: 0.5556.
60 minus 55 is 5.
Squared, 25.
Over 55: 0.4545.
50 minus 45 is 5.
Squared, 25.
Over 45: 0.5556.
50 minus 55 is −5.
Squared, 25.
Over 55: 0.4545.
5 Add them
0.5556 plus 0.4545 plus 0.5556 plus 0.4545 is 2.0202.
6 The cut-off
df is rows minus 1, times columns minus 1: 1 times 1 is 1.
Careful!
Not the number of cells minus 1.
Rows minus 1, times columns minus 1: 1.
Chi-square is always a right-tail test: big means far from what we expected.
Row 1, column 0.05.
Row 1, column .05: the cut-off is 3.841.
7 Decide
The chi-square curve for df 1.
Past the cut-off, 3.841, is the reject part.
2.0202 is not past the cut-off.
So we do not reject H zero.
8 Say it in words
In words: at the 5% level, there is not enough evidence that the group and the answer are not independent.
This does not prove H zero is true.
We just cannot rule it out.
9 Check
Let's check: the expected counts give the same row totals.
Men: 45 plus 55 is 100.
Women: 45 plus 55 is 100.
So it is right!
Let's look back.
First: write the hypotheses.
Then: totals.
Then: expected counts.
Then: each part.
Then: add them.
Then: the cut-off.
Then: decide.
Then: say it in words.
Last: check.