Step by step
The question
Question: we saw the counts 12, 18, 30, 40, and expected 10%, 20%, 30%, 40%: at the 5% level, do the data fit?
1 Write the hypotheses
H₀ is the "nothing new" idea.
Here: the data fit the expected parts.
Hₐ is what we want to show.
Here: the data do not fit.
2 Expected counts
If H zero is true, each cell expects n times its part.
A: 100 times 0.1 is 10.
B: 100 times 0.2 is 20.
C: 100 times 0.3 is 30.
D: 100 times 0.4 is 40.
Every expected count is at least 5, so the test works.
3 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 10 is 2.
Squared, 4.
Over 10: 0.4.
18 minus 20 is −2.
Squared, 4.
Over 20: 0.2.
30 minus 30 is 0.
Squared, 0.
Over 30: 0.
40 minus 40 is 0.
Squared, 0.
Over 40: 0.
Careful!
We always divide by the expected count, not the observed one.
4 Add them
0.4 plus 0.2 plus 0 plus 0 is 0.6.
5 The cut-off
df is the number of cells minus 1: 4 minus 1 is 3.
Chi-square is always a right-tail test: big means far from what we expected.
Row 3, column 0.05.
Row 3, column .05: the cut-off is 7.815.
6 Decide
The chi-square curve for df 3.
Past the cut-off, 7.815, is the reject part.
0.6 is not past the cut-off.
So we do not reject H zero.
7 Say it in words
In words: at the 5% level, there is not enough evidence that the data do not fit.
This does not prove H zero is true.
We just cannot rule it out.
8 Check
Let's check: the expected counts add up to 100, like the observed ones.
And the differences O minus E add up to 0.
So it is right!
Let's look back.
First: write the hypotheses.
Then: expected counts.
Then: each part.
Then: add them.
Then: the cut-off.
Then: decide.
Then: say it in words.
Last: check.