Step by step
The question
Question: weekday: 60 early, 40 late; weekend: 40 early, 60 late: 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: weekday, 100; weekend, 100.
Each column's total: early, 100; late, 100.
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.
Weekday, early: 100 times 100, over 200, is 50.
Weekday, late: 100 times 100, over 200, is 50.
Weekend, early: 100 times 100, over 200, is 50.
Weekend, late: 100 times 100, over 200, is 50.
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.
60 minus 50 is 10.
Squared, 100.
Over 50: 2.
40 minus 50 is −10.
Squared, 100.
Over 50: 2.
40 minus 50 is −10.
Squared, 100.
Over 50: 2.
60 minus 50 is 10.
Squared, 100.
Over 50: 2.
5 Add them
2 plus 2 plus 2 plus 2 is 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.
8 is past the cut-off.
So we reject H zero.
8 Say it in words
In words: at the 1% level, there is enough evidence that the group and the answer are not independent.
9 Check
Let's check: the expected counts give the same row totals.
Weekday: 50 plus 50 is 100.
Weekend: 50 plus 50 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.