Step by step
The question
Question: north: 45 yes, 55 no; south: 30 yes, 70 no: at the 1% 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: north, 100; south, 100.
Each column's total: yes, 75; no, 125.
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.
North, yes: 100 times 75, over 200, is 37.5.
North, no: 100 times 125, over 200, is 62.5.
South, yes: 100 times 75, over 200, is 37.5.
South, no: 100 times 125, over 200, is 62.5.
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.
45 minus 37.5 is 7.5.
Squared, 56.25.
Over 37.5: 1.5.
55 minus 62.5 is −7.5.
Squared, 56.25.
Over 62.5: 0.9.
30 minus 37.5 is −7.5.
Squared, 56.25.
Over 37.5: 1.5.
70 minus 62.5 is 7.5.
Squared, 56.25.
Over 62.5: 0.9.
5 Add them
1.5 plus 0.9 plus 1.5 plus 0.9 is 4.8.
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 0.01.
Row 1, column .01: the cut-off is 6.635.
7 Decide
The chi-square curve for df 1.
Past the cut-off, 6.635, is the reject part.
4.8 is not past the cut-off.
So we do not reject H zero.
8 Say it in words
In words: at the 1% 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.
North: 37.5 plus 62.5 is 100.
South: 37.5 plus 62.5 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.