Step by step
The question
Question: east: 50 agree, 50 disagree; west: 65 agree, 35 disagree: 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: east, 100; west, 100.
Each column's total: agree, 115; disagree, 85.
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.
East, agree: 100 times 115, over 200, is 57.5.
East, disagree: 100 times 85, over 200, is 42.5.
West, agree: 100 times 115, over 200, is 57.5.
West, disagree: 100 times 85, over 200, is 42.5.
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.
50 minus 57.5 is −7.5.
Squared, 56.25.
Over 57.5: 0.9783.
50 minus 42.5 is 7.5.
Squared, 56.25.
Over 42.5: 1.3235.
65 minus 57.5 is 7.5.
Squared, 56.25.
Over 57.5: 0.9783.
35 minus 42.5 is −7.5.
Squared, 56.25.
Over 42.5: 1.3235.
5 Add them
0.9783 plus 1.3235 plus 0.9783 plus 1.3235 is 4.6036.
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.
4.6036 is past the cut-off.
So we reject H zero.
8 Say it in words
In words: at the 5% 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.
East: 57.5 plus 42.5 is 100.
West: 57.5 plus 42.5 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.