Step by step
The question
Question: small: 12 low, 18 high; medium: 20 low, 20 high; large: 28 low, 12 high: 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: small, 30; medium, 40; large, 40.
Each column's total: low, 60; high, 50.
And everyone: 110.
3 Expected counts
If there is no link, each cell expects its row total times its column total, over the grand total.
Small, low: 30 times 60, over 110, is about 16.3636.
Small, high: 30 times 50, over 110, is about 13.6364.
Medium, low: 40 times 60, over 110, is about 21.8182.
Medium, high: 40 times 50, over 110, is about 18.1818.
Large, low: 40 times 60, over 110, is about 21.8182.
Large, high: 40 times 50, over 110, is about 18.1818.
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.
12 minus 16.3636 is −4.3636.
Squared, 19.041.
Over 16.3636: 1.1636.
18 minus 13.6364 is 4.3636.
Squared, 19.041.
Over 13.6364: 1.3963.
20 minus 21.8182 is −1.8182.
Squared, 3.3059.
Over 21.8182: 0.1515.
20 minus 18.1818 is 1.8182.
Squared, 3.3059.
Over 18.1818: 0.1818.
28 minus 21.8182 is 6.1818.
Squared, 38.2147.
Over 21.8182: 1.7515.
12 minus 18.1818 is −6.1818.
Squared, 38.2147.
Over 18.1818: 2.1018.
5 Add them
1.1636 plus 1.3963 plus 0.1515 plus 0.1818 plus 1.7515 plus 2.1018 is 6.7465.
6 The cut-off
df is rows minus 1, times columns minus 1: 2 times 1 is 2.
Careful!
Not the number of cells minus 1.
Rows minus 1, times columns minus 1: 2.
Chi-square is always a right-tail test: big means far from what we expected.
Row 2, column 0.01.
Row 2, column .01: the cut-off is 9.210.
7 Decide
The chi-square curve for df 2.
Past the cut-off, 9.210, is the reject part.
6.7465 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.
Small: 16.3636 plus 13.6364 is 30.
Medium: 21.8182 plus 18.1818 is 40.
Large: 21.8182 plus 18.1818 is 40.
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.