Step by step
The question
Question: junior: 30 bus, 20 car, 10 walk; senior: 20 bus, 35 car, 15 walk: 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: junior, 60; senior, 70.
Each column's total: bus, 50; car, 55; walk, 25.
And everyone: 130.
3 Expected counts
If there is no link, each cell expects its row total times its column total, over the grand total.
Junior, bus: 60 times 50, over 130, is about 23.0769.
Junior, car: 60 times 55, over 130, is about 25.3846.
Junior, walk: 60 times 25, over 130, is about 11.5385.
Senior, bus: 70 times 50, over 130, is about 26.9231.
Senior, car: 70 times 55, over 130, is about 29.6154.
Senior, walk: 70 times 25, over 130, is about 13.4615.
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.
30 minus 23.0769 is 6.9231.
Squared, 47.9293.
Over 23.0769: 2.0769.
20 minus 25.3846 is −5.3846.
Squared, 28.9939.
Over 25.3846: 1.1422.
10 minus 11.5385 is −1.5385.
Squared, 2.367.
Over 11.5385: 0.2051.
20 minus 26.9231 is −6.9231.
Squared, 47.9293.
Over 26.9231: 1.7802.
35 minus 29.6154 is 5.3846.
Squared, 28.9939.
Over 29.6154: 0.979.
15 minus 13.4615 is 1.5385.
Squared, 2.367.
Over 13.4615: 0.1758.
5 Add them
2.0769 plus 1.1422 plus 0.2051 plus 1.7802 plus 0.979 plus 0.1758 is 6.3592.
6 The cut-off
df is rows minus 1, times columns minus 1: 1 times 2 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 0.05.
Row 2, column .05: the cut-off is 5.991.
7 Decide
The chi-square curve for df 2.
Past the cut-off, 5.991, is the reject part.
6.3592 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.
Junior: 23.0769 plus 25.3846 plus 11.5385 is 60.
Senior: 26.9231 plus 29.6154 plus 13.4615 is 70.
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.