Step by step
The question
Question: online: 60 happy, 40 unhappy; shop: 45 happy, 55 unhappy: at the 2.5% 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: online, 100; shop, 100.
Each column's total: happy, 105; unhappy, 95.
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.
Online, happy: 100 times 105, over 200, is 52.5.
Online, unhappy: 100 times 95, over 200, is 47.5.
Shop, happy: 100 times 105, over 200, is 52.5.
Shop, unhappy: 100 times 95, over 200, is 47.5.
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.
60 minus 52.5 is 7.5.
Squared, 56.25.
Over 52.5: 1.0714.
40 minus 47.5 is −7.5.
Squared, 56.25.
Over 47.5: 1.1842.
45 minus 52.5 is −7.5.
Squared, 56.25.
Over 52.5: 1.0714.
55 minus 47.5 is 7.5.
Squared, 56.25.
Over 47.5: 1.1842.
5 Add them
1.0714 plus 1.1842 plus 1.0714 plus 1.1842 is 4.5112.
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 0.025.
Row 1, column .025: the cut-off is 5.024.
7 Decide
The chi-square curve for df 1.
Past the cut-off, 5.024, is the reject part.
4.5112 is not past the cut-off.
So we do not reject H zero.
8 Say it in words
In words: at the 2.5% 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.
Online: 52.5 plus 47.5 is 100.
Shop: 52.5 plus 47.5 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.