Step by step
The question
Question: we saw the counts 15, 25, 35, 45, 30, and expected equal parts: at the 2.5% 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.
Each cell: 150 times 1/5 is 30.
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.
15 minus 30 is −15.
Squared, 225.
Over 30: 7.5.
25 minus 30 is −5.
Squared, 25.
Over 30: 0.8333.
35 minus 30 is 5.
Squared, 25.
Over 30: 0.8333.
45 minus 30 is 15.
Squared, 225.
Over 30: 7.5.
30 minus 30 is 0.
Squared, 0.
Over 30: 0.
Careful!
We always divide by the expected count, not the observed one.
4 Add them
7.5 plus 0.8333 plus 0.8333 plus 7.5 plus 0 is 16.6666.
5 The cut-off
df is the number of cells minus 1: 5 minus 1 is 4.
Chi-square is always a right-tail test: big means far from what we expected.
Row 4, column 0.025.
Row 4, column .025: the cut-off is 11.143.
6 Decide
The chi-square curve for df 4.
Past the cut-off, 11.143, is the reject part.
16.6666 is past the cut-off.
So we reject H zero.
7 Say it in words
In words: at the 2.5% level, there is enough evidence that the data do not fit.
8 Check
Let's check: the expected counts add up to 150, 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.