Step by step
The question
Question: plan: 32 better, 18 same; none: 20 better, 30 same: 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: plan, 50; none, 50.
Each column's total: better, 52; same, 48.
And everyone: 100.
3 Expected counts
If there is no link, each cell expects its row total times its column total, over the grand total.
Plan, better: 50 times 52, over 100, is 26.
Plan, same: 50 times 48, over 100, is 24.
None, better: 50 times 52, over 100, is 26.
None, same: 50 times 48, over 100, is 24.
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.
32 minus 26 is 6.
Squared, 36.
Over 26: 1.3846.
18 minus 24 is −6.
Squared, 36.
Over 24: 1.5.
20 minus 26 is −6.
Squared, 36.
Over 26: 1.3846.
30 minus 24 is 6.
Squared, 36.
Over 24: 1.5.
5 Add them
1.3846 plus 1.5 plus 1.3846 plus 1.5 is 5.7692.
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.
5.7692 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.
Plan: 26 plus 24 is 50.
None: 26 plus 24 is 50.
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.