Step by step
The question
Question: boys: 10 math, 15 art, 25 sport; girls: 15 math, 20 art, 15 sport: at the 10% 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: boys, 50; girls, 50.
Each column's total: math, 25; art, 35; sport, 40.
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.
Boys, math: 50 times 25, over 100, is 12.5.
Boys, art: 50 times 35, over 100, is 17.5.
Boys, sport: 50 times 40, over 100, is 20.
Girls, math: 50 times 25, over 100, is 12.5.
Girls, art: 50 times 35, over 100, is 17.5.
Girls, sport: 50 times 40, over 100, is 20.
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.
10 minus 12.5 is −2.5.
Squared, 6.25.
Over 12.5: 0.5.
15 minus 17.5 is −2.5.
Squared, 6.25.
Over 17.5: 0.3571.
25 minus 20 is 5.
Squared, 25.
Over 20: 1.25.
15 minus 12.5 is 2.5.
Squared, 6.25.
Over 12.5: 0.5.
20 minus 17.5 is 2.5.
Squared, 6.25.
Over 17.5: 0.3571.
15 minus 20 is −5.
Squared, 25.
Over 20: 1.25.
5 Add them
0.5 plus 0.3571 plus 1.25 plus 0.5 plus 0.3571 plus 1.25 is 4.2142.
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 0.1.
Row 2, column .10: the cut-off is 4.605.
7 Decide
The chi-square curve for df 2.
Past the cut-off, 4.605, is the reject part.
4.2142 is not past the cut-off.
So we do not reject H zero.
8 Say it in words
In words: at the 10% 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.
Boys: 12.5 plus 17.5 plus 20 is 50.
Girls: 12.5 plus 17.5 plus 20 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.