Step by step
The question
Question: we saw the counts 160, 25, 15, and expected 70%, 20%, 10%: 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: 200 times 0.7 is 140.
B: 200 times 0.2 is 40.
C: 200 times 0.1 is 20.
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.
160 minus 140 is 20.
Squared, 400.
Over 140: 2.8571.
25 minus 40 is −15.
Squared, 225.
Over 40: 5.625.
15 minus 20 is −5.
Squared, 25.
Over 20: 1.25.
Careful!
We always divide by the expected count, not the observed one.
4 Add them
2.8571 plus 5.625 plus 1.25 is 9.7321.
5 The cut-off
df is the number of cells minus 1: 3 minus 1 is 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.
6 Decide
The chi-square curve for df 2.
Past the cut-off, 5.991, is the reject part.
9.7321 is past the cut-off.
So we reject H zero.
7 Say it in words
In words: at the 5% level, there is enough evidence that the data do not fit.
8 Check
Let's check: the expected counts add up to 200, 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.